Good evening aspirants. Welcome to the daily news analysis by Shankara A. Sakadami for the date 17th October 2021. These are the list of articles we will be discussing today. We will be discussing five articles today. In that three will be from today's newspaper. One article will be from 16th October newspaper and another article will be from 14th October newspaper. We will be having two problems questioned and three main questions today. So stay till the end of the discussion to get the full impacts of the discussion. Without wasting much time let us get into the today's news analysis section. Let us start today's discussion with this article. This is an open article taken from the 14th October newspaper. This article gains relevance in relation to the recently published global hunger index. Through this article the author proposes a solution to the rampant malnutrition prevalent in the country. This article is a well structured one. This is how we must write our main censor. See the introduction part of this article. Here the author gives two nutrition related data. First he quotes the fifth national family health survey here. This survey has covered 17 states of our country and indicated that the nutrition related indicators have worsened continuously. To substantiate his claim of nutrition deficiency in India further he quotes comprehensive national nutrition survey 2016-18. This survey too highlights the micronutrient malnutrition among Indians. See this is how you must write your introduction. It must have three features that is it must be relevant to the question. It must be to the point and crisp and it must have some valid data most preferably from a government organization. When the introduction addresses these three points your introduction will be more attractive and will help you fetch more marks. Now moving on to the article. Let us see the views of the author regarding the nutrition security in our country. To help provide a better nutrition security the author gives various steps that must be taken. Let us go over this one by one. First is improving dietary diversity. This is mainly done to address malnutrition in our country. Next is promoting kitchen gardens for decentralized vegetable production and addressing the nutrition deficiency problem locally. Next is reducing post harvest losses which will in turn make the food mode affordable. We can also make programs like PDS more nutrition sensitive. See currently our PDS program tries to address only the hunger part and not the malnutrition part. So more focus must be given to addressing the micronutrient deficiency. Next important feature to ensure nutrition security is focusing on human empowerment. See human empowerment is not an end in itself. It has very many positive effects like by focusing on human empowerment we can lower TFR that is the total fertility rate. We can also provide better nourishment to the children and focusing on human empowerment will also help in providing better nutrition security. This is what the author is trying to address. Next comes nutrition education and using digital technology to identify target groups and better manage the resources in ensuring nutrition security. Other measures that the author suggests to ensure nutrition security includes the enforcement of standards and regulation and improving water sanitation and hygiene. See here the author has given an all-incompassing solution to address the nutrition deficiency holistically. All these points will be helpful in your main sensor. From this article you can have an idea how we must have a more aerial view of all social problems. See UPSC mainly in the mains GS part expects from us a more wider understanding of a problem rather than a more deeper view. To help you understand this better see how the author incorporated digital technology enforcement of standards as a solution for addressing nutrition deficiency. This is what I mean when I say we must have a wider view. Hope you guys are getting the point that I am trying to deliver here. No moving on. Later in the article the author focuses on the agricultural food system to address nutrition deficiency. See for the population to eat better we must sow better that is we must produce good quality crops and to help the population in switching to a healthy dietary pattern we must focus on shifting the agricultural production pattern as well. Currently we are following monocropping of rice and wheat that must be replaced with mixed cropping. This will ensure diversity in dietary pattern and it would also be a good for the environment. Here in this part the author dives deep into the issues with the agricultural food systems that hinder in achieving the goal of nutrition security. So before looking at the problems in India's agricultural food system let us see some facts about the India's agriculture sector. See India produces sufficient food, feed and fiber to sustain about 18% of the world's population. Agriculture contributes about 16.5% of India's GDP and employs 42.3% of the workforce but still India faces small nutrition rate. So there must be some problems in the agricultural food system. According to the author there are four problems facing the agricultural food system of our country. These problems are economic sustainability, ecological sustainability, poor adoption of agricultural technologies and the extreme vulnerability to climate events which is becoming more frequent with the rapid climate change. So we saw the problems that the Indian agri-food system is facing. The author also proposes some solutions to the problems. So to address the problems the author suggests that the long-term goal of the agriculture must not be to enhance farm incomes but also to ensure better access to safe and nutritious foods. Additionally the agricultural food system needs to be reoriented to minimize the cost on environment and climate. So having discussed about the measures that need to be taken to ensure nutrition security and the challenges the agri-food system faces, let us discuss a little bit about World Food Day. See the World Food Day is celebrated on 16th of October every year. World Food Day marks the establishment of food and agricultural organization. The theme of World Food Day 2021 is our actions are our future. Better production, better nutrition, a better environment and a better life. See these four betters that is the better production, better nutrition, better environment and better life represent the food and agricultural organizations contribution to the sustainable development goals. See FAO that is the Food and Agricultural Organization was established in 1948 and have been an active participant with India. This organization has helped India mainstreaming agro-biodiversity, greening agriculture, promoting nutrition-sensitive agriculture and strengthening national food security. In January this year that is 2021, FAO in collaboration with Nithya Yoke and the Ministry of Agriculture convened a national dialogue to evolve a framework for transition to a more sustainable agricultural food system by 2030 and identify pathways for enhancing farmer's income and achieving nutritional security. This helped India in both enhancing our farmer's income and to make our agricultural system more resilient. To conclude a sustainable agricultural food system must have the following features. First is sufficient nutritious and safe foods must be made available to all people at a affordable price. Second is nobody must be hungry or suffer from any form of malnutrition. Third is food wastage must be minimal and the food supply chain must be more resilient to shocks. Fourth is food system must help combat environmental degradation or climate change. So when a agricultural food system has the above mentioned features the sustainable agricultural food system can deliver food security and nutrition security for all without compromising the economic, social and the environmental basis. So in this segment we discussed various aspects of nutrition security how by evolving a sustainable agro food system we can address nutrition deficiency we also saw how to write a good mains introduction and how having a bird's eye view will help us write a better mains answer. Hope it helped everyone. So let us conclude this discussion and move on to the next article. Now look at this editorial article that appeared on yesterday's newspaper. See very recently on October 13th our Prime Minister Narendra Modi launched Gatti Shakti national master plan.  To put it in simple words Gatti Shakti is a digital platform which will bring 16 ministries together for integrated planning and coordinated implementation of infrastructure connectivity projects. See as an umbrella integrator of 111 lakh crore worth of projects under national infrastructure pipeline for 2020 to 25 this initiative aims to appeal to the national integration. The syllabus covered by this article is highlighted here for your reference. Please go through it. We will go through the Gatti Shakti national master plan in more detail a bit later before that let us briefly go through the national infrastructure pipeline. See the key to any successful economy is a robust infrastructure with an equally strong supply chain. This is also true for our country which has a goal to become a 5 trillion dollar economy by 2025. The national infrastructure pipeline is a step in that direction. See the national infrastructure pipeline is a initiative that will provide world-class infrastructure across the country in order to improve the overall quality of life for all citizens. The initiative will provide improved project preparation and attract both domestic and foreign direct investment in our economy. That is its main aim is to improve the project preparation and attract investments into infrastructure sector. The national infrastructure pipeline has been made on the best effort basis by putting together informations provided by various stakeholders including various ministries, departments, state governments and private sectors across infrastructure sector as identified in the harmonized master list of infrastructure. The national infrastructure pipeline will fulfill all the crucial factors that will help India in achieving its target of becoming a 5 trillion dollar economy by the financial year 2025. It will cover both economic and the social infrastructure projects as well. So now coming back to the Gatti Shakti national master plan. See the infrastructure creation in India has suffered for decades from multiple issues. There was a lack of coordination between different departments. For example, once a road was constructed, other agencies dug up the constructed road again for activities like laying of underground cables, gas, pipelines, etc. This not only caused great inconvenience but was also a wasteful expenditure. To address this, efforts were put in place to increase the coordination so that all cables and pipelines and etc. could be laid simultaneously while the road was being laid. This is the core feature of Gatti Shakti national master plan. See the Gatti Shakti national master plan will address all these past issues through institutionalizing holistic planning for all the stakeholders for all major infrastructure projects. See the focus here is on holistic planning instead of planning and designing separately in compartments with no real-time sharing of information. The project will be designed and executed with a common vision that is a holistic vision. It will incorporate the infrastructure schemes of various ministries and state governments like Bharatmala, Sagarmala, inland waterways, dry or land ports, Udaan, etc. and economic zones like textile clusters, pharmaceutical clusters, defense corridors, electronic parks, industrial corridors, fishing clusters, agri zones will be covered to improve connectivity and make India more business competitive. To put it in simple words, Gatti Shakti national master plan will aid concerned ministries or departments to prioritize connectivity enhancement for ensuring last mail connectivity to economic zones in different time frames. In collaboration with Bhaskar Acharya national institute for space application and geoinformatics, the development of a GIS based enterprise resource planning system will enable all stakeholders and the network planning group which consists of infrastructure connectivity ministries to participate in spatial planning, evidence based decision making, administration and effective monitoring of the plan on a periodic and real-time basis. The portal which has over 200 layers will enable visibility of all essential network linkages and assist national planners in making better decision in the logistic sector. So, we have a holistic understanding about what this Gatti Shakti national master plan is about. Now, let us see what are the positive sides of the plan. First is it will help in reducing wasteful expenditure. See, due to lack of planning and coordination, there was wasteful expenditure in the past like we discussed So, through this plan, there will be holistic planning so the wasteful expenditure can be minimized. Second is it will help lower the cost of logistics as a percentage of GDP. Presently, the cost of logistic as a percentage of GDP is at 14 percent. When properly imprimatined, this can be brought down to 8 percent of GDP. Third is it will attract major investments which in turn will boost job generation, which in turn will increase demand for goods and commodities. Understand this, when the transportation cost decreases, the cost of goods and commodities will also decrease. This will induce demand in the market and indirectly boost the protection which will again contribute to GDP. The fourth and the most significant positive of this plan is land acquisition. See, most of the projects in India are stuck due to land acquisition delays. So, in this the government is using GIS that is the geographic information system to identify waste and polluted land with no prospect for agriculture and develop these lands as industrial parks. Now, having gone through the positive sides of the Kartishatni National Master Plan, let's see some challenges that this plan might face. The first challenge is environmental. See, there is a challenge for reducing vehicular emission from the road freight growth in order to meet the climate change commitments. Apart from this, keeping input prices in check due to exorbitant diesel tax also poses a challenge. The second challenge is the role of active center state partnership for infrastructure building. See, since states have an important role in all this, center can convince the states and states in their part can ensure port connectivity, land availability for motorways, trains, industrial clusters and corridors and rely on a political agreement and active collaboration. Though this center state partnership might act as a hindrance through active communication between the center and state, this challenge could be overcome in the future. So, in this section of the discussion, we discussed about the national infrastructure pipeline, the newly launched Kartishatni National Master Plan, its salient futures, its advantages and the challenges it might face. With this, let us end this discussion and move on to the next article. Now, for our next discussion today, I have chosen this FAQ article. As we know, the cold crisis is continuously making headlines in recent times. So, in this discussion, we are going to learn about the ongoing cold crisis which our country is facing and we are going to learn about the intensity of the issue and then we will move on to the factors that led to the crisis and we will end our discussion with a brief note on the way forward. That is how we are going to address the issue. Now, with this idea in mind, let us get into our discussion. The syllabus covered by this FAQ article is highlighted below for your friends. Please go through it. See, as of now, the amount of cold held in India's thermal power plants are going down to a very low level. In fact, many power plants are operating with nearly zero stock reserves or the stocks that could last only a very few days. So, basically, this shortage has put India in the edge of a power crisis and now the intensity of this issue has started to gain ground because already some Indian states are witnessing partial load shedding in order to save power. That is, in some states, the power companies have already began to reduce the electricity consumption by switching of the power supply to customers since the entire system is at risk. Note here that this is happening at a time when there is a broader energy crisis across the world. That is, a sharp price is witnessed in the prices of natural gas, coal and oil in the international market. So, I hope this gives you a brief background of the issue at hand. Now, let's briefly see about the current status of coal in India. See, when you take India, around 70% of our power needs need to be met through coal and on that line, the coal India limited supplies over 80% of the total coal consumed in our country. Note that the government usually mandates the power plants to hold the stock that could last for at least two weeks. But the crisis has made it almost impossible because according to the data released by Central Electricity Authority, the average coal stock in 135 thermal power plants in India is capable of lasting for just four days. And of that 135 thermal power plants, 112 thermal power plants are currently operating with stocks that are very critical or at an extremely critical level. Due to this, the government has reduced this stock requirement. That is, from 14 days of stock requirement to just 10 days. And the main intention behind this is to avoid hoarding and also to ensure a more equitable distribution of coal among the power plants. So, by now we have a broad picture of the ongoing coal crisis that is faced by our country. Now, let us move on to see some factors that have contributed to this ongoing crisis. See, the first and foremost reason for the current crisis is the un-energetic or the unexciting domestic production of coal. And according to the BP Global Energy Statistics, the domestic coal production is said to be stagnated since the year 2018. During 2018, the total coal production in India peaked at 12.8 Exa Joules. See, we know that Joules is a unit of measurement of energy and here Exa means 10 to the power of 18. So, in 2018, when the coal production reached its peak, we produced 2.80 Exa Joules worth of coal. Now, moving on to the second reason for the current crisis. For the past three years, the coal imports have been continuously declining. That is, the amount of coal imported from other countries to meet domestic demand has also come down. For example, the import of coal has dropped from a peak of 6.46 Exa Joules in 2016 to about 4.22 Exa Joules in the year 2020. If you notice here carefully, despite the shortage or stagnation even in previous years, the coal crisis has begun to gain intensity only at the present time. One major reason for this is because last year we had our economy shutdown to tackle the COVID-19 pandemic, because of which there was not much power usage. But then when the situation changed and when we came back to our normal lives, the demand for power had once again resumed. So, now the effect of coal supply and demand mismatch has manifested into the present crisis that we are facing. Adding to this, there exist some deeper structural problems that have plagued the power industry in general as well. For instance, it is said that the price paid for power is not equal to the total production costs. Similarly, as pointed out in the news article, the revenue of distribution companies for the financial year 2019 covered only about 70% of their total costs. This situation here is counter-intuitive and such issues have led to the decrease in private investment in the power generation. Despite there is a raising demand for power, see the number of people added to the middle class is continuously increasing. So, the power consumption, that is the demand for power also is continuously increasing. But still then the private investment is decreasing in the power production and the power distribution sectors. Another important factors that led to the present crisis is the monopoly in the mining of raw material by power sector companies like coal India limiters which are in run for profit. More importantly, there is not much financial incentive available for the major producers across the supply chain to ramp up production. Apart from these, other factors have led to the insufficient supply of coal including short-term issues like flooding in coal mine areas, transportation issues, labor disruption in coal mining countries and also sudden raise in power demand as the economy revived back from the pandemic. So, we saw about the brief background of the current issue and the major factors that led to the current issue. Now, let us see the way forward. That is, let us see some solution that will help in addressing the present issue. See at present, countries are trying to cut down their fossil fuel consumption. But despite that, India being one of the top two consumers of the coal in the world is planning to further increase the production of fossil fuels. But to power a country that is as huge as India, it is important for India to ramp up its production in order to meet the raising coal demand and also cut down the country's reliance on imported coal. But for short term, it is good to ease restriction on imported coal as well to overcome the crisis for a temporary period. Also, a fundamental pricing reform in both coal production and in the power sector is the need of the HAR to make investment in coal production and thermal energy production more lucrative. Let us recap a little bit. Here we discussed about the present coal crisis in India, what are the reasons that have led to the present situation and some measures that can be taken to address the issue. So, with this, let us wind up this news article and move on to our next article. Now, let us take up this news article. The article here reports about the Adavaleshwara temple which is situated in the Vilipuram district of Tamil Nadu. Due to lack of proper maintenance, this temple is in a bad condition and vegetation has grown on the mandabam over the sanctum sanctorum. Due to this, the temple is leaking inside during rains and its structure is also deteriorating. So, on that line, the Hindu religious and the charitable endowments have been requested to take necessary steps to preserve the temple. So, this is the basic crux of the news article here. Now, based on this context, let us learn about some important points about the Adavaleshwara temple. See, this Adavaleshwara temple is a Hindu temple which is dedicated to Lord Shiva. The presiding deity here is Adavaleshwara and the mother of the deity is Periyanayaki. As per the inscription found here, the temple is believed to be more than 1300 years old. As per the sources, Kulothunga Chola has contributed to the development of this temple which was already famous back then. Following him, later, the Kulothunga Chola II has also contributed even more to the temple and made it huge and grand. Note that there are about 52 inscriptions in the temple and they all majorly talks about the contribution made to the temple by various kings, especially the Chola kings. And this list includes Rajendra Chola, Kulothunga Chola I, II and III and also Raja Raja Chola. One among the inscriptions also talks about the Kidaram Kundat Chola in Sali, where food was regularly served. And note that it belonged to the period of Rajendra I. In fact, the earliest inscription found on the wall of the temple relates to the 11th year of Raja Kesari Varman, who is identified with Kulothunga II. Kulothunga II here is praised as Pu Mevu Valar. Keep in mind, the temple is a rare south-facing Shiva temple dating from the Chola period because most of the Chola temples are north-facing. Now, coming to the structural aspects of the temple. See, the temple has a Balipidam, Javasthambam and Nandi situated in a raised platform. And this platform is built in a way that it is facing the sanctum located outside the temple. The temple has also got a tank situated on the right side of the temple. And when you take the sanctum sanctorum of the temple, it consists of a sanctum, antarala, arthamandabam, 16 pillard muhamandabam, a mahamandabam and also a front mukhamandabam. It has a sanctum, antarala, arthamandabam, 16 pillard muhamandabam, a mahamandabam and also a front mukhamandabam. So, these are the important facts related to Adavaleshwara temple. So, with these facts in mind, let us move on to the next article for our discussion. Now, let us take this news article. This article says that after many delays in its upgrading process, the army air defense expects to make significant progress in terms of acquisition and trail in the coming months. According to the defense authorities, these include additional indigenous akash surface-to-air systems, the underdevelopment medium range surface-to-air missile systems and the Russia's IGLA-S short-range air defense system. That is, V-Sharad systems. See, air defense functions at three levels. First is the gun missile system. Second is the medium range system. Third is the high range system. Within this, air defense guns are of two types. They are air defense gun missile system and the air defense self-propelled guns. The army is looking for guns in both these categories. IGLA-S very short-range air defense system comes under this category. In the medium segment, army has the indigenous akash surface-to-air missile, while the medium range surface-to-air missile, that is MRSM, fits in the high range. So, in this context let us briefly know about Russia's IGLA-S very short-range air defense system in the preliminary point of view. Know that short-range air defense is a group of anti-aircraft weapons and tactics that have to do with defense against low-altitude air threats like helicopters and low-flying close air support aircrafts. IGLA-S man-portable air defense system is a Russian origin very short-range air defense system. Make a note here, it is a man-portable surface-to-air missile. This system is the latest variant of IGLA man-pads developed before this. The IGLA-S version has a larger range than the previous variants as well as it has sensitive seekers and effective resistance to new defensive measures. It is designed for use against visible aerial targets at very short range such as tactical aircrafts, helicopters, unmanned aerial vehicles that is UAVs, cruise missiles, head-on or receding in the presence of natural background clutters and countermeasures. Talking about its specifications, see the missile is powered by a solid fuel rocket motor engine. It has contact and grazing fuse detonation mechanism. See, a fuse is a part of the device that initiates function. In some applications such as torpedoes, a fuse may be identified by function as the exploder and a contact fuse is nothing but a fuse that is placed in the nose of the bomb so that it will detonate or explode on contact with the hot surface. This new variant has a operational range of six kilometer altitude of three kilometers along with all weather capability. So horizontally it will cover six kilometers and vertically that is altitude it will cover three kilometers. The maximum peak speed of the missile is about Mach 1.6. For guidance purposes it uses a dual waveband infrared guidance system that is it has a infrared seeker and a guidance system which gives it the capability to lock into infrared light waves given off by the engine of the aircraft. As per the manufacturer this system is having a heat probability of 0.8 to 0.9 that is 9 out of 10 launches will be successful. See whenever you come across a new military equipment you have to concentrate on some aspects for the plumes. So here in case of iglias we must give more attention to its type which is v-sharad that is very short range surface to air different system then comes the range. So in this case it is six kilometer horizontally and altitude it will cover three kilometers. Then we must focus on how it is launched. Here iglias is launched using a man portable model okay and it can also reach a speed of Mach 1.9. Additionally this missile like we discussed has a infrared seeker and a guidance system with a heat probability of 0.9. This is a general template. This template can be used whenever you see a military hardware make news okay. By using this template you can cover any news article holistically for plumes okay. So with this let us conclude the news analysis segment and move on to the practice plumes question section. We have two practice plumes question today. Let's take up the first question. Which of the following statement is not correct about adavaleshwara temple. First is it is dedicated to larchiva. Second statement it is a rare south facing temple from the chola period. Third statement karigala cholan has contributed to the development of this temple. Fourth statement its inscription talk about the kidaram konda cholan salai. See based on our discussion we can infer that the second statement given here is incorrect because as I said earlier kulottunga cholan has contributed to the development of this temple which was already famous and following him kulottunga chola 2 contributed even more to the temple and made it huge and grand therefore the right answer here is option c since the question wants us to identify the wrong answer only so now moving on to the next question it is a two statement question first statement is the akash is a mid-range air-to-air missile built by the India state-owned defense research and development organization. Second statement the akash system can employ multiple air targets while operating in fully autonomous mode they have asked us to find the incorrect statements okay here the statement one is wrong because akash is a mid-range surface-to-air missile not a air-to-air missile this missile is built by India's state-owned defense research and development organization so statement one is wrong see it is a anti-aircraft missile and it can employ multiple air targets while operating in fully autonomous mode so the second statement is correct so the correct answer for this question is option a one only because the question asks for the wrong statement the main question based on today's discussion is displayed here we have three main questions today okay go through the questions completely first question will be based on the recently announced gathisakthi plan second question will be based on the nutrition security and the third is based on the current coal crisis okay go through the question completely and write your answers following the structure it must have a pattern starting it must have an introduction then you must address the question and it must have a good conclusion it must it can be a recommendation or a way forward so that's all for the discussion today if you like this video do like and comment and share it with your friends and subscribe to Shankara IS Academy YouTube channel thank you


The mystery of high-energy cosmic rays, presented by Science at NASA. Hold out your hand for 10 seconds. A dozen electrons and muons just zipped unfelt through your palm. The ghostly particles are what scientists call secondary cosmic rays, subatomic debris from collisions between molecules high in Earth's atmosphere and high-energy cosmic rays from outer space. This daily shower, which never stops, is a sign of violent events in deep space. It's believed the majority of cosmic rays come from supernova explosions. When massive stars explode, they blast most of their material into space. The expanding shock waves can break apart interstellar atoms and accelerate the debris to unimaginably high energies. Other unknown cataclysmic phenomena may be at work too, especially for the most energetic cosmic rays. Un Suk Saab, a professor of physics at the University of Maryland says, cosmic ray particles with energies as high as 10 to the 20th power electron volts have been measured on the ground. This is more energy than we have achieved in the most powerful man-made particle accelerators. But how do natural cosmic ray accelerators pump so much energy into these particles? This is one of the biggest mysteries in astrophysics. While indications of the energies of cosmic ray particles can be measured from the ground, Saab and colleagues have taken their study to higher elevations, directly measuring particles from space before they break up in Earth's atmosphere. The cosmic ray detector known as CREME, the Cosmic Ray Energetics and Mass Investigation, has been launched to the stratosphere above Antarctica on board long-duration helium-filled balloons. By lofting the detector above 99% of the Earth's atmosphere, researchers get a better idea of what cosmic rays are like before they collide with nuclei in the air above the detector. CREME is able to measure the energy and direction of each incoming cosmic ray particle and identify the particle type by measuring its charge, thereby providing clues to the particle's origin and acceleration mechanisms. Since 2004, the team has flown CREME seven times over Antarctica, accumulating more than 191 days of data from altitudes as high as 120,000 feet. They're about to go even higher. CREME is heading for space. A reconfigured CREME detector is scheduled to travel to the International Space Station in 2017, aboard SpaceX's Dragon spacecraft on a Falcon 9 rocket. Named ISS CREME, it will remain installed on the Japanese experiment module, also known as Kibo, for at least three years. So says the ISS provides an excellent monitoring platform for high-energy cosmic rays. The station allows for long-term monitoring in lieu of multiple limited-duration balloon flights, providing direct unimpeded access to incoming cosmic rays without atmospheric interference. The longer exposure times on the space station allow for the measurement of higher energies. ISS CREME has a goal of measuring the highest energy possible for direct measurement of high-energy cosmic rays. She adds, the mysterious nature of cosmic rays serves as a reminder of just how little we know about our universe. For more high-energy science from Beyond Earth's atmosphere, stay tuned to science.nasa.gov.


Well, welcome to episode 19 on exponential and logarithmic equations. We'll also talk about logarithmic scales in this episode. This is Math 1050 College Algebra, and I'm Dennis Allison. I teach mathematics here at Utah Valley State College. Let's see, in the last episode or two we've been talking about logarithms, and today we're going to look at some of the applications of logarithms, and this will extend into episode 20 as well. To look at the applications that we're going to consider today, I want to remind you of the four fundamental laws of logarithms. The first one is if we can go to the green screen, that if you take the log base B of n plus the log base B of n, you get the log base B of the product mn. What was the second law? Who remembers? Involves a quotient. The log base B of m minus the log base B of n equals the log base B of m over n. We'll be using that one today also. The third law says that if you have a constant multiple of a log base B of m, you get, who can tell me what will that become? Log base B of the quantity m to the c, exactly. I'll put that in parentheses just to set it off. And number four says if you have a, well, let's say if you have B and you raise it to the log base B of m, then m is the result. Okay, now with these four properties, we can do quite a few applications of logarithms. And let's look at the first. In fact, let's look at the list of objectives that we have here. In this episode, we want to look at solving exponential equations. We want to look at how to solve logarithmic equations. And then finally, we'll look at what we mean by logarithmic scales. So we'll finish up with that. Okay, let's go to the first example, to the first graphic of examples. And here we have three exponential equations. Now, surely these are all equations because we have two expressions that equal to each other. There's a variable involved. And so these three equations are referred to as exponential equations.  And some exponential equations require a little work to solve them. For example, let's look at these. In the first one, 7 to the x power is 49. I'll write that one here on the screen. 7 to the x power is 49. Well, what exponent should I put on 7 to get 49? I think x should be 2. So there's really no work involved in solving for x. But we have solved an equation for an unknown. In the second equation. Okay, now in the second example, we have 2 to the x minus 3 equals 1 half. Let's write that on the green screen. 2 to the x minus 3 equals 1 half. Now, this is another example of an exponential function because the variable is in the exponent. Excuse me, an exponential equation. And it's only slightly more complicated than that last one that we just solved. You see, this says 2 to the x minus 3 is equal to 2 to the negative 1. So if I make the bases alike, then the exponent should be alike, which means that x minus 3 equals negative 1. Now, you see, what we've done is actually more subtle than you may first realize. What we've done is we've changed the exponential equation into a linear equation. And we've been solving these sorts of equations since, say, the first year in algebra. x minus 3 is negative 1. And therefore, x is equal to, if I add 3 to both sides, x is equal to 2. So what we do is when we have a problem that looks more complicated, we change it into a different form. Rather than us going to the problem, we change the problem into something that's easier to solve. And in this case, the linear equation is very quick. If I just check that answer right below it, if I substitute in a 2 up here for x, we have 2 to the 2 minus 3 power is 2 to the negative 1 power, and that's equal to 1 half. So that does check. Okay, now, in the third example, we have 5 to the 2 plus 3x equals 5. Okay, now, if I divide both sides by 5... Excuse me, yeah, if I divide... No, I wrote that wrong, didn't I? 5 times e to the 2 plus 3x. Now, if I divide both sides by 5, I have e to the 2 plus 3x is equal to 1. And so what must this exponent be if I want this exponential expression to equal 1? Must be 0. Must be 0. So 2 plus 3x equals 0. What kind of an equation is this? This isn't an exponential equation. It's a linear equation. So you see, once again, we change this into a linear equation. So we're making the problem come to us, you see, by coming up with an equation that's easier to solve. So this says that 3x is equal to negative 2, and therefore, x is equal to negative 2 thirds. And I think if you substitute that back in here, you'll get an exponent of 0, which means 5 times 1 is equal to 5. So that will check out again. Okay, but now we ask the question, what if we're not able to get this down to a linear equation quite so easily? And let's go to the next examples and look at some problems where that would be more difficult. In the first equation here, we have 2 to the x equals 7. And I think you can see the problem is the 2 and 7. There's no common base I can convert those to, whereas before, I could somehow evaluate those fairly quickly. So let's take that first problem, 2 to the x equals 7. What I'm going to do is apply logarithms. I'm going to take a logarithm on both sides. Suppose I take log base 10, or just log on the left, and I'll take log on the right. What this does is it allows me to move the variable down onto the ground so that I can solve for it. And if I bring the x out in front, this is x times the log of 2 equals the log of 7. Now, you know, this is no longer an exponential equation. The variable is not in the exponent. What I have is a number that's a coefficient of x, the log base 2 is the coefficient of x, equals a constant on the other side, log 7. This is a linear equation. It just doesn't look quite as simple as the linear equations that I solved from that last graphic, but I have a multiple of x is equal to a constant. So if I solve for x, this says x is equal to the log of 7 divided by the log of 2. Now, this is what I would refer to as the exact answer. This is the exact answer. But if you want to know approximately how much it is, for example, is this number bigger than 2, is this number bigger than 3, would have to carry out a division using our calculator. So let me just move my calculator in on top of this problem. If you can zoom in on this. Okay, so I'm going to take the log of 7 and I'm going to divide it by the log of 2.  When you divide these two logarithms, you actually get an infinite decimal expansion, and the calculator rounds it off to about 9 decimal places, or 10 significant digits. So we'll say this is about 2.807, about 2.807 for my approximate answer. So going back here to the green screen, x is approximately 2.807, and this is my approximate answer as opposed to the exact answer that was given above. Now on an exam, I may ask you to give me the exact answer and an approximate answer, in this case out to 3 decimal places. So you'd want to show this answer and this answer as well. Now, you notice at the beginning of this problem, I took a log on both sides, and I chose to take the common log on both sides because I had that available on my calculator, and so I was able to make this division. What other logarithm could I have taken right here so that I could have used my calculator? Could have taken a natural log. Okay, now over here on the side, let me just try reworking the same problem using natural logarithms. So we have 2 to the x equals 7. Now what if I were to take the natural log on the left and the natural log on the right? Of course natural log means log base e, and so if I move the x out in front, like I did before, x log 2, equals natural log of 7, and so x is equal to the natural log of 7 over the natural log of 2. This is the exact answer if you're using natural logarithms, and it looks different than that one, but I think when I approximate it, I think I'm going to get the same result. So to figure out what is the approximate answer, let's go back to our calculator, and if you can zoom in on this one more time, I'm going to try dividing the natural log of 7 divided by the natural log of 2. So the natural log of 7 divided by the natural log of 2 is, and I don't know if it shows up so well at home, but I have exactly the same digits, 2.807, I should say, and so we do end up with the same approximate answer, 2.807, and therefore when you give the exact answer, it could be given in either form, and when you give the approximate answer, depending on how many decimal places you want to give, these answers look the same down here. Okay, let's go to the next log, the next exponential equation where we'll have to use logarithms to solve it. This one says 15 to the 2 plus x power equals 1 third to the x power. Okay, let's solve that one right here. 15 to the 2 plus x power equals 1 third raised to the x. Now, you see, we don't have a common base that I can express both of these in, and so I'm going to need to use logarithms for this. So I'm going to try taking the common log on both sides, log of 15 to the 2 plus x power, and on the other side, log of 1 third raised to the x. Now, if I move my exponents out in front, 2 plus x times log 15 equals x log 1 third. Now, it's kind of hard to get used to this, but you see, log 15 is a constant. Just like 2 is a constant, the log of 15 is a constant. We don't see exactly its decimal representation. The variable is x. And on the other side, log of 1 third is a constant and the variable is an x. Now, what I want to do is to solve for x in this linear equation. And the way I solve for x is I need to multiply this out and get all my x's on one side. So I'm going to call this 2 times the log of 15 plus x log 15 equals... Now, what can I do with the log of 1 third? What's another way of expressing the log of 1 third? The log of 3 to the negative 1. The log of 3 to the negative 1. Okay, let's think of this as 3 to the negative 1. And if I bring my exponent out in front, that would be minus x log 3. So what we're doing is we're changing 1 third to 3 to the negative 1. The exponent comes out in front and when I put it in front, I'll just put it in front of the x and make it minus x log 3. Okay, so I have x's in this expression, x log 15, and I have x's in my expression on the right. I want to get these together. So let's get them both on the left-hand side. x log 15 plus x log 3. And the one that has no x's in it, I want to put it on the other side. So I'll subtract 2 log 15. So I get negative 2 log 15. Okay, so all my x's are on one side. I'm going to factor out the x. And I have log 15 plus log 3 in parentheses all multiplied by x equals negative 2 log 15. Okay, so I have a multiple of x equals a constant on the other side. So to solve for x, just like I would in any linear equation, I divide by the coefficient of x. It seems funny to call this the coefficient because it's number one, it's on the right of x. It's not on the left. And also, it's written with logarithms. We tend to think this is a variable, but it's just a constant. So I'm going to divide by that. And I have minus 2 log 15 divided by the log of 15 plus the log of 3. By the way, can anyone think of a way to write this denominator as a single logarithm? See, we have a question, Jenny. Yeah, that's what I was going to wonder. Could you just make that the log of 45? Exactly. See, whenever you add two logs together, you get the log of the product. So it might look simpler to call this negative 2 log 15 over log 45. You know, as a matter of fact, you might go further and say, Dennis, couldn't we get rid of the coefficient negative 2 and put it inside as an exponent? Would that make it look simpler? Well, it might make it look a little shorter, but I'm not sure that having a negative 2 as an exponent is really anything that simplifies it. So I think I'll leave that negative 2 out in front. So I would say this is the exact answer to this problem. Negative 2 log 15 over log 45. If you left your answer in the form we had here before we simplified the denominator, I would accept that as the exact answer for this problem as well. But what is the approximate answer? Let's see, Matt, I notice you're working on your calculator there. Are you evaluating this? Yes. What are you getting for this approximate value? Let's say up to about three decimal places. Negative 1.423. Negative 1.423 is what Matt got. I'm going to divide that on my calculator up here and let's just see how he came up with that answer. So I'm going to put this right on the top of the problem. And if you'll zoom in, I'm going to take negative 2 times the log of 15, and I'm going to divide this by the log of 45. Equals. And there we have negative 1.423. And Matt rounded up to make it 423. Let me do this another way. Using this first answer that I came up with, I'm going to take negative 2 log 15. But now I'm going to make a mistake here and I want someone to tell me what the mistake is I'm going to make. If the camera can zoom out for just a moment, let's look at this answer that's right below the calculator. I'm going to enter negative 2 log 15 divided by log 15 plus log 3. So here's how I'm going to do it. If you can zoom now back in on the calculator, I'm going to say divided by log 15 plus log 3. Now, when I push enter, I'm not going to get the same answer. And now I get negative 0.954 something. What did I do wrong when I entered that? We should have put parentheses around my divisor. See, I want to divide by the sum of the log of 15. Oh, and also, I notice I put a times, not a plus there. That was just a typo in my part. There's a multiplication, not a plus. So that's also a mistake. So let's go back and do this correctly. This is negative 2 log 15. Divided by the quantity, and I'm going to open parentheses. There's a little button here that shows open parentheses. And now log 15 plus log 3. Close parentheses. And now we should get the same answer that Matt got just a moment ago and that we got here in class. And there it is, negative 1.423 when we round off. So the reason I point this out is because students, I think, sometimes forget to put in the parentheses when they're doing a division by a sum. The other mistake I made is I just hit a wrong key and I hit a multiplication up there instead of a plus, but that was my own fault. Not something you would probably be doing at home. Okay, so we have two different exact answers. Either form would be okay. And then we have one approximate answer written out to three decimal places. Okay, let's go back to the graph. You can look at the third problem that asks us to solve an exponential equation. This says e to the 2x minus e to the x minus 6 is 0. Let's write this one on our green screen. E to the... Well, let's see, for some reason my marker isn't working. Let me pick up another marker down here. Try it again. e to the 2x minus e to the x minus 6 is 0. What are some of the things that makes this problem look different from the others that we've just been working? We have two different terms with x's in it. So this is still an exponential equation, but we have the x in exponents in two different terms. We have more terms. We have three terms here instead of two terms that we had before. And you know what I'm thinking is this one looks a lot like a quadratic equation. It's an exponential equation, but it looks quite a bit like a quadratic equation. I'm thinking maybe we could factor this into e to the x and e to the x. By the way, what's the product of e to the x times e to the x? e to the 2x. Yeah, let's just write that out. e to the x times e to the x. The rule says you add the exponents and you get e to the 2x. So this product will give me e to the 2x. Now, if I factor this like a quadratic expression, this is in what's called a quadratic form, I think I should put a negative 3 here and a plus 2 over there. Let's check this and see. This product is 2 e to the x. This product is negative 3 e to the x. So when I combine them, I get negative e to the x. And by the way, the last product will give me negative 6. So I think we have been able to factor this, something we didn't see come up in any of the previous problems. And this product is 0. So either e to the x plus 2 is 0 or e to the x minus 3 is 0. Now in this first case, this says that e to the x is negative 2. This is impossible. Because if you look at the graph of e to the x, if you graph the function f of x equals e to the x, let me just remind you what it looks like. Here's the x-axis. My graph is a little small. I hope you can see this at home. If I graph e to the x, you remember the target points were, first of all, to go up one. And if I go to the right one, I should go up e, which is about 2.7. And if I go to the left one, I should go up 1 over e. And to make a long story short, the graph is going to look like this. Now my question is, when does the graph equal negative 2? Here's negative 2 down here. When does that graph come down to negative 2? It never does. So there's no solution. An exponential expression has to give me a positive answer. It can't even give me 0 for the answer. So it's got to give me a positive answer. Certainly not negative 2. So my conclusion from all this is there's no solution to this portion of the equation. But the other side says e to the x could equal 3. So I'm looking for an exponent that I can put on base e to give me 3. I don't see any way to make these two bases look just like, not any obvious way right now. So I'm going to take a logarithm on both sides, and I want to be able to use my calculator. So which logarithm do you suspect we should use here? Natural log, because there's a base e involved in the exponential equation. So I'm going to take the natural log of e to the x, and I'm going to take the natural log of 3. You see, if these two expressions are equal, they should have the same natural logarithm. And so this says x natural log of e is the natural log of 3. And what is the natural log of e? It's 1, so this is just x. So we get x equals ln of 3. This is my only exact answer. This is the only exact answer we get because we didn't get any answer over here on the other side. And if I approximate it, let's say we approximate it to 3 decimal places, I'm going to put my calculator in here if you can zoom in on this. And we want to approximate the natural log of 3, so we'll just enter natural log of 3. And we get about 1.098. And if I round that off, we'll call it 1.099. So that's my approximate answer. We'll call that 1.099 right there. So back to the green board, 1.099 is the approximate answer given to 3 decimal places. Okay, so once again, we have an exact answer and we have an approximate answer. The exact answer, because there's nothing better than the exact answer, but when we ask how big is that number, for example, is that more than one? Is it more than one and a fourth? Is it more than one and a half? It's hard to say until we approximate it. So both answers are important that we be able to calculate. Okay, I'm going to take one more exponential equation and I want someone to tell me how to graph this. I'm going to keep the camera on the green screen. What about this problem? Suppose I have 2 times 9 to the x plus 3 equals 18. Now do I need logarithms to solve this problem? I don't think so. It may be difficult to tell in the beginning, but I think there's a way that I can make these have the same bases, and I don't have to use logarithms to solve it. What would you do first to solve this problem, Matt? Divide by 2, yeah, and so now we have 9 to the x plus 3 equals 9. Yeah, so now what? Well, then I know that 9 raised to the 1 equals 9. So I would find something that would make the exponent 1. So what would the exponent have to be? Negative 2. Okay, actually Matt's a little bit ahead of us. If there's a first power over here, x plus 3 is going to be 1. So now here's our linear equation, and then when I solve for x, x is negative 2. You were exactly right. So in this case I don't have to use logarithms, but if I erase this and change one number, what if I change this number not to 18 but to 20? Now here's a problem where I have to use logarithms to solve it. So you see it's very easy to go from the relatively simple problems to the ones that are a bit more complex. In this case, we'd still divide by 2. 9 to the x plus 3 power is equal to 10, but the problem is I no longer have a common base between these two. So at this point we have to shift gears, go to logarithms, and just to be different, what if I take a natural log on both sides? Even though there's a 10 in the problem, let's take a natural log. Natural log of 9 to the x plus 3 equals the natural log of 10. I'll bring the x plus 3 out in front times the natural log of 9 equals the natural log of 10. Then I'll multiply this out so that I get x times the natural log of 9 plus 3 times the natural log of 9 equals the natural log of 10. And there's only one term with an x in it, so I want to isolate that. So what I'll do is subtract 3 natural log of 9 from both sides, and I have x natural log 9 equals the natural log of 10 minus 3 natural log of 9. Therefore x is equal to, now this is the exact answer, so I can still put an equal sign, the natural log of 10 minus 3 times the natural log, oops, natural log of 9, over the natural log of 9. This is a perfectly good answer, although it is possible to maybe change this a bit, make it look a little bit more compact. Jenny? Why couldn't you divide when you have x plus 3 in parentheses and then you have the natural log I think you had of 9? Why can't you just divide from there that natural log of 9? Actually you could, that would be an alternative. What Jenny is suggesting is we could have isolated the x perhaps faster if rather than multiplying this out if we had just divided by the natural log of 9, call this ln of 10 over ln of 9 equals x plus 3. I think that probably would have been a little quicker. And that would be equivalent to this. You see, what you would get is if you divide this out you'd have the natural log of 10 over the natural log of 9 minus 3. And that's the answer that you would arrive at up there. I think Jenny is exactly right, that would have been a little bit shorter. To get the approximate answer, I don't think I'll do this in the interest of time, but what you'd want to do is think of parentheses around all this. When you enter this in your calculator, open parentheses natural log 10 minus 3 natural log 9, close parentheses, and then divide by the natural log of 9. Okay, that would give you your approximation. Okay, well, I think we've covered fairly thoroughly now exponential equations. Let's look at logarithmic equations. And we have another graphic now with several problems of a different variety than what we've just seen. Now these three problems are equations because we have expressions that are set equal to each other and there's an unknown that we'd like to figure out. But rather than having exponentials, we have logarithms, the variables are inside logarithms. So we call these logarithmic equations and we solve these using the same rules, the same four basic laws of logarithms that we've been applying already. And later in this episode and in the next episode we'll see applications where equations like this come up. So we're just preparing for those by finding out how to solve them now. In the first equation we have the log base 3 of x plus 1 equals 4. Let's solve that one right here. The log base 3 of x plus 1 equals 4. Now I would call this a logarithmic equation because the variable is inside of a logarithm. Anyone have a suggestion on how we could solve this? Form an exponential equation from this. Yeah, change this to exponential 4 because we're used to looking at exponential statements much more than we are looking at logarithmic statements. Let's see, Matt, what would be the exponential version of this? 3 raised to the 4 equals x plus 1. x plus 1, yes, because all of that is inside the logarithm. This is sometimes referred to as the argument. So x plus 1 is the argument, so x plus 1 is equal to the exponential quantity. So this says that 81 is equal to x plus 1. By the way, what sort of equations are we solving here? Linear. These are linear equations, yes, exactly. So once again, we've made the problem come to us by changing it to a form that we like better, linear equation and linear equation. Now it's fairly simple to get out of this. x is just equal to 80. So that's our solution. And by the way, this is the exact solution and there's no approximation, of course, involved in this. So that's the only answer we'd have to give. Okay, now in the other two examples, the problems get a little bit more complicated. Let's take part B. The log of x minus the log of x plus 2 is equal to 2. I'll write that one on the green screen. The log of x minus the log of the quantity x plus 2 is equal to 2. Well, this one's a bit more complicated than the last problem because it's a logarithmic equation, but I have two different logarithms involved in both of them. How could I somehow get a single logarithmic expression here? David, what would you do? How could you get a single logarithm there? Can we pull the log x out anyway? Well, we can't factor the log out, but we do have a difference of logs. Now one of our properties tells us a way to reduce that. Jeff, what are you thinking of? We could change that to the log of x over x plus 2. Right, yes. So that was our second property of logarithms that says if you have the log of m minus the log of n, it's the log of m over n. So now I have one logarithm, and I have a more complicated expression inside, but it's only one logarithm. Now, you might ask the question, Dennis, which one of these is simpler? Well, in one sense, the first one is simpler because I had simpler arguments, simpler logarithmic expressions. This one's simpler because there's only one logarithm. So it's sort of in the eyes of the beholder, which one is simpler. But for our purposes, I think we like this one better because the next thing I'm going to do is change this to exponential form. Let's see, Ginny, what is the exponential version of this logarithmic equation? It would be 10 raised to the 2 is equal to x over x plus 2. Equals x over x plus 2. Okay, now this is no longer a logarithmic equation because we have no logarithms. If anything would call this a rational equation, because I have a ratio of polynomials, I have a rational expression. So this is a rational equation. We haven't seen one of those in this episode, whereas up above, I had a logarithmic equation there and there. So once again, I've changed it to a form that I find easier to solve. You might say, gee, is that easier to solve? Yes, I think it is because look, x over x plus 2 is equal to x over x plus 2.  What do you do? Both sides by x plus 2? Yeah, I'm going to multiply both sides by x plus 2. So this is 100 times x plus 2 and I'll multiply on the right by x plus 2 and of course they cancel there, so I have x. Now, what kind of an equation is this? This is an irrational equation. This is a linear equation. So the problem just keeps changing. This is a linear equation now, but this is the one that we'll actually choose to solve. So we had a logarithmic equation, a rational equation, a linear equation, and now I think we're ready to resolve all this. We have 100x plus 200 is equal to x. So that says that 99x is equal to negative 200. And that says that x, I'll put it over here on the side, x must be negative 200 over 99. This is my exact answer. Now, if on your calculator you were to divide this out, this would be an infinite repeating decimal and your calculator wouldn't be able to display an infinite decimal because it's not infinitely wide. So this is the exact answer. You wouldn't want to give this as an in an approximate form unless I ask you to give it as an approximate. So our final answer is negative 200 over 99. I don't think there's anything that'll reduce there, so that's the final result. Okay, once again I took a logarithmic equation. At this time I made it a rational equation and then we got a linear equation out of that. Okay, in the last example for logarithmic equations, we have the natural log of x plus the natural log of x plus 6 equals 2 times the natural log of x plus 4. I'll write that one here. We have the natural log of x plus the natural log of the quantity x plus 6 equals 2 times the natural log of the quantity x plus 4. Now what I've tried to do in these examples is to find various problems that illustrate different ideas and how to go about solving these sorts of problems. This one is a little different from either of the previous two logarithmic equations. Now, I'm going to begin on the same note. I'm going to combine these two logarithms into a single log. And if I add two logarithms, I get the natural log of the product. So this is the natural log of the quantity x times x plus 6. Equals, and over here I have a coefficient. Where's another place we could hide the coefficient? As an exponent to the x plus 4. Bring it in as an exponent. So this is the natural log of x plus 4 squared. Okay, now what makes this problem different is this time, I'm going to say, well, I've got the natural log of this number equals the natural log of this number. Well, if I have the same natural logarithm values, the two numbers inside must be the same. So I'm going to set this product equal to x plus 4 squared. So I have x times x plus 6 equals x plus 4 squared. So I'm thinking if the two natural logs are the same, the two arguments must have been the same from the beginning. So this time I just eliminate the natural logs. Of course, when I multiply out the left-hand side, I'm going to get an x squared. And when I square the right-hand side, I'm going to get an x squared. So it looks like this should be a quadratic equation. It's an equation, and I have some x squared terms involved. So let's multiply this out. We have x squared plus 6x. And let's see, what is x plus 4 squared? Plus 8x plus 16. Plus 8x plus 16.  So we're down to 6x equals 8x plus 16. Now, this is actually now a linear equation. So here we are, changing from one form to another until we get something that we would like to solve. If I collect all my x's on the right, I have 2x. And if I subtract off the 16, minus 16, and so I get x equals negative 8. x equals negative 8. Unfortunately, you know what? It's sort of sad. I'm going to have to throw that answer out. There's no solution for this problem. Does anyone know why? Because it'll leave you with a negative natural log, and it never is. Yeah, exactly. You see, if I take this number and substitute it back in up here, I'm going to have the natural log of a negative 8. The natural log of negative 8. Over here, I'll have the natural log of negative 2. And here I'll have the natural log of negative 4. Well, you can't take natural logs, or any logs, of negative numbers. Let me show you why. I'll just move that answer up here to the top. And I want to make myself some space. So we thought the answer was negative 8, but then we had to throw it out. Here's the reason. If I graph the function f of x equals ln of x. I'm choosing ln because these are all natural logarithms. And if I graph that function, you remember this is the inverse function for e to the x. So when I graph the natural log function, I'm going to steal the target points from the function e to the x. But I'm going to move them sideways. So starting at the origin, I go over 1. And if I go up 1, I go over e, about 2.7. And if I go down 1, I go over 1 over e, or about a third, roughly. And so my graph looks like this. Now, looking at this graph, what would you say is the natural log at 1? What's the natural log at 1? 0. Is 0. Yeah, the function of that is 0. This is e right here. What's the natural log of e? 1. Is 1. If I were to go over here to 3, just to take a wild guess, could anyone just, based on this graph, what would you say is the natural log of 3? 2, possibly? Yeah, it's actually a little bit less than 2. But since my graph isn't just perfect, it's kind of hard to tell. It's actually a little bit less than 2. You could find it on your calculator. But let me ask you this. What's the natural log of negative 8? Here's negative 8 out here. Well, let's see, there's no graph out here. So there's nothing for me to locate it, to locate a function value. As a matter of fact, negative 8 is not in the domain. If I were to take this graph and press it onto the x-axis, the domain would only be the positive numbers, not negative numbers. That's why I threw out negative 8. Although in the algebra I came out with a negative 8, it's what's referred to as an extraneous root in mathematics. That is, it's an extra root, but it doesn't really solve the problem. And that's because I can't put in negative 8 there. I can't put in negative 8 there. And by gosh, I can't even put in negative 8 over there. Can't take the natural log of negative 4. So it's a number that has to be thrown out. There was no solution for the third part, part C of that example. Okay, so we have to be aware that sometimes extraneous roots come up. And you might ask, you might wonder, how is it that negative 8 could have been introduced? If negative 8 wasn't a solution in the beginning, how did we ever get to negative 8 later on? Well, in the beginning when I wrote that natural log equation, there was a subtle assumption in there that the x's couldn't be negative or even 0. But when I changed it to a quadratic equation and then a linear equation, that assumption was dropped out and we were looking for every possible solution. So negative 8 may be a solution to the quadratic and to the linear equation, but it's not a solution to the natural log equation because of the restriction on the domain there. Okay, so now you know a little bit about solving exponential and logarithmic equations, but really the only way you learn how to do these accurately is to do the homework problems, try solving these on your own. And you may make a few mistakes, but you learn from your mistakes and I think you'll get good at it. Okay, let's change topics. And I want to show you another article that I pulled out of a Salt Lake Tribune just recently. I think I showed you several articles that represented exponential problems. Here's an article that I think could use some logarithmic application. So if you can show the green screen, I'll just show you this came from the Daybreak section back on May the 1st, 2002. So I've had this for quite a while. It's from the Food section and the title was called Mile to Wild. And now if you go further down, go further down here, the article is actually down here at the very bottom of the page and the article has to deal with measuring the heat in various peppers and how hot some peppers can get. Is there a way to actually classify them mathematically how hot a pepper can be? So they discuss in the article something referred to as the Schofield heat scale right here and I won't go into all the details of scientifically how you measure the heat index of a pepper, but it's the graph that I'd like to show you. Now this may not show up very well on the camera, but the scale goes from zero at the bottom up to 600,000, 600,000 what they call Schofield units. Schofield is the name of the person who invented this technique for measuring the heat of a chili pepper. Well they start off at zero and at zero we have a yellow bell pepper. Now frankly I don't remember anyone who's jumped up from the table and run out of the room looking for a glass of water because they bit into a yellow pepper. I mean yellow peppers just don't have any heat to them. I don't know about green peppers and those red peppers, those bell peppers, but I think they're all classified as being rather low. Now if you go up a little bit higher you may recognize this anaheim pepper and its heat index varies from 500 units to 2,500 units and there's a little line that goes down here very close to zero. So what they're saying is there's not that much heat in an anaheim pepper and so on the scale you can hardly tell it from zero. But the next one above it, have to follow the line up to it, is the serrano pepper and the serrano pepper has 8,000 to 22,000 scofield units. Why do you think it varies from 8,000 to 22,000? Stages of its growth. Stages of its growth. Yeah, what could be other factors that would affect the heat index of the pepper? Personal taste? Well, I don't know if personal taste enters into this but it could be just the environment it was grown in whether it got more water, less water, what sort of nutrients it got. So probably you could find one serrano pepper a little hotter than another but they say generally they vary from 8,000 to 22,000 units. So if we put that on the graph, look, it's way down here near the bottom too because that first marker is at 50,000 units. So these peppers are clumped together. Now the next one to come to is the Thai pepper. Now I've had experience with Thai peppers and to me they are hot. Now for some people they may say, no, Dennis, you're such a wimp but I find Thai peppers to be really hot and Thai peppers vary from 50,000 to 100,000 Schofield units and here on the scale, well they haven't marked off at 100,000, at 100,000 but look, the scale goes on up to 600,000. That's because we haven't gotten to the really biggie here the red savina habanero pepper and its Schofield unit measure is 350,000 to 577,000. Okay, Stephen, oh, yeah, there so you can see that. Stephen's nodding his head, yes, he's bit into those. Stephen, how did you find them? Tasty or? Well, it stays in your mouth for about three hours afterwards and it's very hot.  I don't think I've ever had one of those but here's the problem that I posed to you. When I draw these on the scale or when the newspaper drew them on the scale this one is way up above all the others and we have three of them clumped together near the bottom. So what would be another way that I could represent this so that they weren't so spread out? So let me suggest an alternative way of representing them. So if you take a look at this graphic that I've prepared I have just picked a number in the range for each pepper. For the Anaheim pepper I set about 2,000. For the Serrano pepper about 15,000. You get Serrano peppers in Mexican food quite a bit. Thai peppers, of course in Thai food you might bite into a Thai pepper and for me it was an experience I haven't forgotten. My eyes watered everything. But then here's the red Savina Habanero pepper and I called it 400,000. The range was a little bit broader than that obviously but here's how the newspaper has shown these. The Anaheim pepper was at 2,000. So I've marked off a horizontal scale 0, 100,000, 200,000, 300,000 on up and so the Anaheim pepper is going to be right here. I'll just put N-A-N for Anaheim. By the way there was the yellow bell pepper right at 0. It would be right on top of that. Then the Serrano pepper at 15,000, okay if that's 100,000, 50,000 15,000 would be right about here. I'll put S-E-R for the Serrano pepper and then there's the Thai pepper at 75,000. Well 75,000 is right about here and there's the Thai pepper and then there's the red Savina habanero pepper at 400,000. Now you see it's distorted way up above. I'll just say red there for that. So these things are not clustered together at all. So here's an alternative. Suppose I were to label my scale 0, 1, 2, 3. I'll try to write it a little bigger so you can see it there. 4, 5, 6 and what I'm going to do is take the logarithm of each of these values up here. So let me just slide this down so you can see what I'm going to do. I'm going to take the log of 2,000. I'm going to take the log of 15,000 and I'm going to take the log of 75,000. And maybe you could help me out here. Those students in the classroom, what is the log of 2,000? 3.30 3.30 approximately. Now how would you have known that was going to be a little bit larger than 3? 3 zeros in the number. 3 zeros in the number. You see the log of 1,000 is 3. So the log of 2,000 should be a little bit more than that. Now for the log of 15,000 this should also be 3. something. It's 4.17. Oh, 4.17. Oh, you're right. 4. 4.17. 4.17. And the reason it's a little bit bigger than 4 is because the log of 10,000 is 4 and this is a little bit more than 10,000. The log of 75,000. Well, this is not up to 100,000 yet so it's not 5. It's going to be 4.875. Okay, and now let's go to the red Savina Habanero. The log of 400,000. Let's see, that's one, two, three, four, five. The log of 100,000 is five. So this is 5. what? 6.02. 5.602. Now on this scale, I'm going to locate them by their logarithms. And so the Anaheim would be at 3.3. Okay, here's Anaheim. The Serrano pepper is 4.17. So that would be right about here, the Serrano pepper. The Thai pepper is about 4.875. That would be right about here. There's the Thai pepper. And the red Savina Habanero is 5.6. 5.6 is right about here. Now my point is this. Look at the difference. These peppers are now rather equally spaced and they're drawn together. Up above they were spread way out. And sometimes when you have data that is widely dispersed, in order to look at it more conveniently, you take logarithms. And this was referred to as a logarithmic scale. And now you can see them closer together. But these spaces, although they appear to be about even, represent ever-increasing spaces back on the real scale.  Here, they're all about the same. Now let me show you how this is applied in science. Can we go to the next graphic? Okay, here are three examples of how logarithmic scales are used in science. First of all, in chemistry, when you measure the pH of a solution, the pH is calculated by taking the negative log of the hydrogen ion concentration that's abbreviated by a square brackets H+. Now, I'm not a chemist and I won't try to explain how you measure the hydrogen ion concentration, but that H+, tells you generally how many free hydrogen ions are floating around in the solution per mole. And if this number turns out to be less than seven, it's said to be acetic. And if this number is greater than seven, it's said to be a base.  Well, those numbers are so diverse that what you do is you take the logarithm of those and then you put a negative on there to make it positive. Let me just give you an example. Suppose the hydrogen ion concentration, H+, could vary between, let's say, 10 to the negative nine up to 10 to the negative two. I'm not sure that those are actually accurate numbers, but it's something like that. You see, this is 0.0000001. I've got nine decimal places there. The hydrogen ion concentration, let's say, is generally bigger than that, but less than or equal to, what if this number is a decimal? 0.01. 0.01, yeah, exactly. Well, you see, those numbers are so long that it's actually nicer to take the logarithm of 10 to the negative nine and the logarithm of 10 to the negative two and then inside here we have the logarithm of the hydrogen ion concentration. But what is the log base 10 of 10 to the negative nine? Negative nine, yeah. Remember, what's the exponent you put on base 10 to get 10 to the negative nine? It's negative nine. And what is the logarithm base 10 of 10 to the negative two? Negative two. So this logarithm would vary from negative nine to negative two. Now, what they do in chemistry to make these numbers positive is they just multiply through by a negative and this is referred to as the pH. That's called the pH. So it's done in an artificial manner. We take a logarithm so that we make these messy decimals into rather nice numbers and we put a negative in front of the logarithm to make the negative numbers into positive numbers and we say if the pH, which is what this is defined to be, if the pH is less than seven, we say that the solution is an acid. And if the pH is greater than seven, we say that it's a base. And if I'm not mistaken, the pH of ordinary water is seven. So if the pH varies from what typical water would be, it's either an acid or a base because it's more or less than seven. Well, you know, I think we have time for about one example before this episode ends. So let's take whole milk and the pH for whole milk is generally about 6.5. Now, that number is less than seven so that means milk is slightly acetic. So it's a lactic acid. So it's only slightly acetic. And what if we wanted to find the hydrogen ion concentration in milk? Well, if I substitute 6.5 into my equation, 6.5 then is negative log hydrogen ion concentration. Before I can solve for the hydrogen ion concentration, I better move the negative over. So negative 6.5 is the log of the hydrogen ion concentration. So changing this to exponential form based in to the negative 6.5 power is the hydrogen ion concentration. And if you multiply that out, you get approximately 0.0000003. So that's roughly the hydrogen ion concentration. We'll continue talking about logarithmic scales in the next episode. Hey, we got it right.