YouTube, what's up? Playoff promo. Oh my god. Oh my god, new packs, new cards, new re-rolls. Oh my... Oh my... All the cards suck.  Champ Baley's fast, but I like pick artists and he doesn't get a good ability. You pick artists, no other ability's good. So Champ Baley's really not that good. The other cards, all trash. Be honest. Okay, the new season card? Gronk is amazing. I'll have him in damn... March. I can't wait to use him in March. Whippity-doo-dah-hoo-ha-hey. Okay, Steve Young. Is Steve Young good, boys? What you're gonna find out right here, right now? You tuned in to see if Steve Young is good. The new Steve Young. Ooh. Now, for me, left hand, right hand, never really mattered.  So that never really mattered. So some people care about that. But Steve Young, he is mobile. He is fast. We're gonna put the same abilities I've been using on the quarterback since Elway came out. Which are what, boys? You two, put them in the comments if you know the abilities. Come on now. Everybody asks me. I don't change it often. Fearless, set feet lead, hot route master. If you guys are new here, hot route master allows you to have all the hot routes at your disposal when you're like Peyton Manning out there. So you have all the hot routes. Set feet lead gives you a little zip on the football when you throw it. And fearless makes it so you never get a pressure on accurate. Now, there's a lot of people out there saying you don't need fearless. That's fine. If you don't want to use fearless, you're at the mercy of EA giving you the hook. And one of my goals when I boot up and play Madden is, you're going to get the hook. But at the end of the day, you have to do whatever you can to avoid the hook. All right. And if you walk out there without fearless, you're kind of opening up a little bit for more of the hook. All right. So that's why I will always use fearless. You know, nothing else is a guarantee in this game. What you need on the field is guaranteed abilities. That's what I love. What are the guaranteed abilities you said? Fearless, pick artist. Set feet lead. We'll zip it in there. Hot route master. Get the ones that are automatic. Do not mess up. The RNG does not get you. So those are guaranteed locked right there. So that's why I say let's take a look at my team before I add Steve Young. Here it is. We still got Elway. We're going to throw the same abilities on Steve Young and give them a try out. CD lamb, zero AP, uh, short and elite. Plexigal burst, one AP, short and elite. Herman Moore, one AP, short and elite. George Kittle, zero AP, short and elite. Munoz, one AP, secure, secure protector. Creed Humphrey has the free identifier. Joe Thomas, two AP for threat detector and edge protector. Threat detector, edge protector combo. That's why he's there.  The one thing I learned is never throw a running back pass without high balling it.  I don't know if I don't know what he's too big to turn around and catch the football. I don't know. Defense, we've made some changes. We've made some changes. You guys asked me, we've made some changes. Last intro I told you guys I've had an inside shade of my safeties. No more. I'm tired of the shade, tired of it. You know why? It lets me down.    When I spend six, seven AP on a corner and he gets beat, what happens?  I'm too old.   I have four DBs. If you guys know, if you guys haven't been living on the rock, I run spinner 98% of the time. So I have four corners in coverage. Rod Woodson, Denzel Ward, Antonio Cremati, and Mike Haynes. Okay. On the outside guys, I have pick artists and I have deep, deep knockout for streaks, for C routes, for just please God help me be anywhere near the ball and knock it out. All right. Just see you want to give yourself the flukey so you get deep knockout and pick artists. I have a strong pick artist belief that everybody on the field will have it or they won't play. That's why I don't have some champ bellies and other cards like that. But in a safety, we're going to rock the same thing except with our deep knockout, we're going to slap on a medium knockout. That way we have, hear it now boys, four deep knockouts, two medium knockouts for the corner routes, posts and slants and stuff like that. And we also have inside stuff on warren sap, which is a super body ability. I'll probably change that, but you know, I'm tired of the Tom Dick and Harry's running inside zone on me. So for me, a little bit tough. Now someone in my chat told me, why don't I use short out elite? It beats man press. Now, once again, there's another man that's been living under a rock. I run tight a lot. So that guy's not out short out. You know, like that's why it's called short out elite. Now I thought about, let me go ahead and I'll use Deebo Samuel. I used a lot and he was a beast. He had both in and out for one, much like feeling much like T.Y. Hill in those cards. But honestly, I didn't love it or not. I'd rather have the big body, even though eggs are kind of got patched. If you guys have noticed, they, they secretly nerfed eggs, but I would rather have the big body of Plexigal Burst, but you might be right. Maybe you guys are having success with T.Y. Hilton. I just think he's a little bit small pause, but I still like the big bodies out there. That's why I went Plexigal rather than use a T.Y. Hilton with short out elite. So there's your answer to that question. That's my team. Let me know what you guys think about the jerseys. We're switching them by every game. What's your favorite jerseys? And again, that's what I know. What jerseys do you want to see me wear in the next gameplay? Until now, take a look at Steve Young and is he worth it? And also, man, Hey, if you're on a budget, you know what, get whatever you want. All my opinions are based on you having a lot of damn money. Okay. And if you don't, and you don't have John Elway, God bless Steve Young, Tom Brady, all these guys are great options.  That's what it's about. Put something on the field and go have fun. Okay. All my opinions are based on you can get any card in the game. So I'm sorry for the budget guys, get whatever fits, whatever works, whatever abilities you like. Boom. But check out Steve Young, see how he does. He's already been out for one day and he's going to be under 300k. So definitely budget card. Let's see. Well, sports really took my all Navy charges jerseys. Wow, that's kind of foul. I mean, I swear to God, where's my that's as good release y'all said I got. I mean, dude, what the just happened on that plate, dude, just got to be trolling. Like the game just has to be trolling bag there. God possession called just hide just rat catch that W. Oh my God. Dude, I was like the fastest little out route ever. I knew that was like my mismatch. Come on, boys. Like God, like, yo, you I swear you can't do that. Of course, I take the man off in command. And I like, oh my God, W. Just the same play every play. Just run the same defense that bagged the four plays in a row. I just feel like bro, playing on the server is such a L on passes like that though. Hendricks tried to guard this. I'm done. I'm high balling it every time now. I learned my lesson. No way we didn't get our feet in on that play. Come on, man.  Like, come on, dude. Pick this off, man. Come on. I got lucky there at all because he blocked his tight end. So my my safety went to a deep blue and instead of man coverage. I mean, I just got to do more slants, man. Like it sounds like so body, but like telling you man, he's just scrambling is just not a part of this game. Simply you got to hit that pass or you're out. Thank you. It's a good job. 56 you got to nobody. You got up there and nobody you block and nobody. I mean, how do you not block this dog? Like he just ran by, he ran by three O linemen and just nobody even put a hand on him. Get a holding call or something. Oh, get there. Just so on offense, man. Oh, free form. You really sent out the running back. God, man. Just too much. Just doing too much. Come on. Thank you, bro. Good man up, W. Good block, Haynes. Good block, Haynes. Good man up. Come on. Like you had him a little bit. It's a little shady. Just high balled it and that's the animation I got command like a dog. Oh, my God, dude. I'm like, dude. Oh, my God. It's just weird. It's just weird as right there. Nothing was clean on that. Like nothing. I'm about done with this. Steve Young card. I'll tell you that man. Oh, LT. Come on, man. He's just so hot. I just had to do that bag. That was probably the first time I actually tried to block right there slide block my running back on the other side. And, you know, that worked out really well right there. That worked out great. See, like, bro, I could score right there. Here we are inside the 10. Let's go buyers. That's why I got you big. Let's go, man. Pick. Oh, come on, man. Dog. Like, come on, man. That's such a bail out. Let's go, man. Another pop up in the air. Can I get a pick six? Can I get a D line pick six? Somebody?  Like, are you not a medium? Let's go, man. Pick. Thank you, man. Time I call one of these. I'm telling you, no matter how fast the quarterback is, boys, y'all not scrambling. Like, we got to get there. Like, we got to realize that it don't matter how fast the quarterback is. You're not scrambling. You're not. The only way to scramble is to step all the way up in the pocket and then get 10 yards. I just like, dude, that was the, I just know that I cannot throw a corner out inside the 30 yard line. I can't ever. I just can't throw a corner out inside the 30 yard line. The corner out, it's because they just run so, when you get inside the 30, the corner vouchers are streaks. Pick. He swatted that, dude. Yo, I didn't need that one. I didn't need that one. I get it again. I feel like you have to go for this. I really feel like you can never punt this ball. Maybe you can. That's all reliable, boys. Are you got tripped up by that? I don't think I can play on conservative. I swear to God. I don't even know what happened to the play I drew up. I don't even know what happened to the play I put on the field. I have no clue. I don't have time to outstand. That's tough. Dude, there's Steve. Yo, Elway coming back. I've seen enough. Oh, I've seen enough, bro. I've seen enough, bro. GG's. All y'all got to do is send the game and we can play. All y'all got to do is send the game. Sports, we can play again. Yeah, we can play again. With TY Hilton? Why? Maybe for short out and short in, but I really don't think it matters. I feel like if you have the abilities, I don't think it really matters too much. Oh my God. Come on, man.  Thank you. Denzel. It's tight end. I swear it's just open, though. That's why I swear to God, dude. I will never not have pick artists on somebody. I thought about, do I put champ with just deep knockout and no pick artists? I feel like anybody makes that play and you can make a play, but they just, they catch picks. When they have an opportunity to catch a pick, it's a pick. There's nothing else about it. And you're all fearless too. Skimbo, you know what they told me today? They said, yo, they patch under pressures. They nerf them. You don't need fearless. Yeah, sir. You don't got fearless. Ah, I could switch. I feel like if I don't high ball that he might pick it though. I mean, dude, I swear Elway throws these passes. I swear to God he does. He might be the only one in the world that can't catch that one, huh? Like, bro, he, you just, speed means nothing, dude.  Like, maybe it does. Oh my God. Like, depressing is just disgusting, man. I have, I don't think I've ever played a man where people just get, just interfered with this much, just absolutely wiped off their routes, absolutely mugged, illegal contact, absolutely just drenched in, I don't know these knockout. Thank you. That's a deep. Oh my God. He's going to catch it. I put him in zone. I put him in zone. So my knockout is gone. Great defense.  You know, we're just going to let him score this. Yeah. And then just bump my people for, I swear to God, this game for real, just fourth down.  The best part about this game is the jerseys. And then we're going, oh my God. Thank God. I swear I had seven touchdowns on the field, didn't I? I stared down one person. I stared down the first person that it was open. Just like the city run the same play over and over and over and over and over. I might get open this play. I might get open this play. Bro, I put the yellows on right. Come on, man. Out of here, bro.  If a three man rush gets pressure like that, you don't have a prayer. You might as well punt on first down. Let's go, Dubby. Good stick. Oh, I got two tight ends in the game. No wonder.    I've never had this much trouble snapping and throwing the ball. Have never had this much problem snapping and throwing the ball ever. Let's go, buyers. Just oh, buyers? The buyers is a horse. Oh my God.  Make him knock the ball out.  Safety. Come on, man. You better start blocking. Oh, look at that speed, bro. Look at that speed. Get the new Steve Young. I can see a world where you just play like this and just put all deep knockouts and just press the people and just have fun. That's what's happening right now. I just have never seen my dudes this big.  These dudes are absolutely strapped. Definitely gonna go there again. Yo, dawg. I had Square open, but he has a Ted Hendricks dude running.   It's first down. Do I need to force the ball to snap? Force the ball. Do I need to do that? Oh, man. Why did we just run right there? I just broke like, man. And the outside dudes getting pressed so God bad that like he almost there to stop that play. I mean, dude, the three men rush is killing me. Snap the ball here, boys. Bro, if he actually like blitzed and guarded or running back, I'd be in shambles. I'd be in absolute shambles if he did that. Nice, man. I don't know why he played zone. Thank God he played zone. Medium knockout. That's Haynes. That's Haynes. Not him. He has, bro. Everybody just got absolutely torched. Everybody got beat so bad like, oh my God. Bro, let me tell you something. I swear to God that was a DDA drive. I swear to God, bro, that was the game saying let's let's let him score this drive. The whole everybody's wide. I swear to God my knockout didn't work on a 12 yard put like swear to God. I believe it 100%. They just let everybody just get wide hoping that drive. I believe it. I believe it 100%, dude.  He watched Henry and said I want to do exactly what he does, except he's not as sharp on offense. He's just not as good on offense. Oh, Steve Young. Look at that speed. Look at that speed. Fumble. He knocked out. Got to be. I just want to do that. We ain't getting no pressure that boy. I should have shade up inside the 30. I know they don't run good corner routes inside the 30. All right. I got to execute right here. Like all the way execute. I mean, he always puts 15 zones on the left side down. He's got to take this side. Let's go. There's just no way he caught all the way back up on it. This is such bad defense.  I need a better. I need a better door right there, Steve Young. We need a better girl. Like we literally need a better girl. Kittle. I bow Jackson. Kittle, both Jackson. Is that where we're at in life right now, Kittle? Get me. Get Steve Young off my field. Get him off the field. Get Steve Young off of my field. I want to quit right now. Get Steve Young out of here. Oh, my God. Elway is it. Oh, my God. He just can't. Oh, he got a good deep release. People are screaming. What are you at? Deep release. What? Push, push, push, push, push, push, push, push. Oh, but he could throw deep though. That'd be cool. Look at him. Look at him.  Fumble. Somebody give me the guy to football. Hell no. I'm baggy. All you got to do down there, man. Shade up. They ain't going to run the corner out the right way. Yo, dog. They all got the same shed animation. Oh, my God. They all got the same shed animation. They all got the exact same shed animation at the same time. My account is 100 percent, but it's 100 percent bugged. I'm convinced next year I'm playing on a different account. Next year we play on a different account. Yeah, I wanted to give myself like a good rep to what to do right here. You know, 36 seconds, no timeouts. He needs a touchdown. I should be able to play decent defense here. I would 100 percent go for it. I would have passed earlier in that drive, honestly. Oh, my God. EA got me with the purple zone instead of deep blue. Oh, my God. Change Steve Young. Yeah, he's out. He's out. GG's.  Steve Young, give away for the chat. You thought the way, how mean you guys were to me tonight. How mean the chat has been to me tonight. You guys want. Yeah, exactly. Hey, yo, what you put on Steve Young?


This video is a bit different in that I'm going to explain lucid dreaming for children, for kids, young people, and you know you can define young people however you want, I mean I'm not going to give you an age bracket but people who are generally young, maybe you're in school or in college or something or maybe you're in primary school, I don't know, or maybe you're a parent and you want to teach your children how to lucid dream. Here are some tips, here's some advice for you. So firstly I want to say and this is probably what you're most concerned about at least according to my comments and emails, you're wondering hold on a second, this lucid dreaming thing, is it actually dangerous and are there any risks associated with lucid dreaming? Well I'll tell you straight away, there are no risks or dangers really of lucid dreaming, the only danger so to speak is that you might want to sleep more often, you might want to spend more time asleep and that's not really a bad thing, if you're young you should be sleeping more because you need more sleep anyway. But in terms of dangers with your mind it's far healthier and better in terms of escapism than any video game you could possibly play. So if you're a parent especially if you're and you're worried about your kid lucid dreaming because he might do some bad things in his lucid dream or in her lucid dream it's far less of a risk than any video game that you could imagine okay and in a lucid dream you're basically, you're going to be drawn to things that are exciting yes but your brain will still have a sort of moral compass anyway, you won't be wanting to go and do crazy things more likely than not in a lucid dream you'll just do kind of fun things you know maybe you'll run around or fly maybe you will climb a big tree climb a skyscraper be a superhero okay it's very unlikely that anything bad or dangerous could or would happen okay so you're trying to lucid dream you're wondering is it different to lucid dream if you're younger well yes but the good news is it's easier it's actually a lot easier to lucid dream if you are young and I'm going to define young and don't get offended by this if you're watching this in your older I would define young as anywhere below the ages of let's say 20 2022 that kind of age that is what I would consider young in this context so if you are under that age you're going to find it a lot easier than most of us certainly you're going to find it easier than you know people above the age of 22 in general there are exceptions of course so good news you'll find it easier now what techniques do you actually need to do in terms of how to lucid dream for children for kids if you're younger there's not really much difference in terms of technique however what you will find is that you you need to rely much less on a physical induction technique than you do to actually just build up the basic habit so this is good news for you you don't need to start you know making crazy alarms on your phone waking up at 3am you know interrupting your sleep I mean you can do yeah you can do it if you want but you will have really good results if you just do the following things meditate every morning for like 10 minutes or so write your dreams down in a kind of like a notepad a journal that kind of thing or on your phone and just generally try and set the intention to lucid dream if you're young your hormones and your sleep cycles and the way that you generally sleep is so good you have no idea how good your sleep cycles usually are that you won't need to put much effort into lucid dreaming to actually have a lucid dream compared to an adult I'm talking about in comparison to let's say a 40 year old 50 year old whatever okay you'll find it so much easier than they will and this is because as we get older we generally tend to get worse quality sleep I mean this is kind of like a big assumption right it really depends on the person but your your REM sleep and your sleep cycles become much more erratic and random the older you get when you're young you normally have these really clean sleep graphs where it goes you know you you jump straight down into light sleep and then medium and then deep sleep and then back up to REM sleep and they sort of go in these really clean patterns like this 90 minute cycles and that's really good for lucid dreaming that's you know the easiest way that's the ideal situation if you want to lucid dream when you're older however you might skip REM sleep for the first two sleep cycles and then you might have like a quick blip of it here and then like it's all over the place in short the good news is you can lucid dream much easier if you're younger so just set the really clear intention that you want to lucid dream tell yourself I will lucid dream tonight and visualize yourself this is kind of the biggest tip I guess visualize yourself becoming lucid later on like in the future you can take a few seconds to do this with me now if you want just imagine you know you're looking down at your hands and you're thinking hold on a second this is a dream I'm lucid right now that's the moment of lucidity okay now you want to visualize yourself doing that looking down at your hands realizing oh this is a dream visualize that situation that scene happening in the future next time you're lucid and this is really powerful just that intention technique alone will probably give you a couple of lucid dreams anyway but if you want to make it even more effective here's what you do so do that throughout the day visualize yourself becoming lucid and then right before you're about to go to sleep in the evening set a really strong intention tell yourself I will lucid dream tonight there's no doubt about it and then visualize just before you go to sleep sort of as you're laying there falling asleep visualize yourself becoming lucid in the future uh so basically that is it if you are a parent um well firstly if you're a kid that's how you do it if you're a parent teaching a kid that's how you teach them it's really simple there's no real danger or different thing you need to do except focus on the intention and the awareness you already have the odds massively stacked in your favor if you're below the ages of 20 to 22 so give it a try and see how it goes


Stand by stand by stand. I just saw the view. It looks like it's completely sunken in I don't know what caused it, but I did see like one of those like mechanical dog a search dog thing Kind of shocking It was an old building. I don't know if they know what happened, but yeah, it's a lot. I Hope everybody's okay, and I hope everybody's surrounding the area is okay But yeah


The mission of the Georgia Council on Economic Education is to help teachers teach economics in the public and independent schools of Georgia. We're very blessed in Georgia that economics and personal finance can be found throughout the curriculum. Every grade level elementary, middle, and high school, including a required high school economics course. When it comes to strategies to teach economics, you're really only bound by your creativity and I suppose the time you have to put those lessons together. The Georgia Council on Economic Education gives you such a head start compared to other content areas that it's unbelievable the opportunities that you have just with their materials that they provide readily at workshops. The Georgia Council on Economic Education helps teachers teach economics by conducting workshops written specifically and created specifically for each grade level in the state of Georgia. Teachers generally come to a full day workshop where they learn some economic concepts. They participate in activities and they actually see and use lesson plans that have been created specifically for them that they can take back to their classroom the very next day and use with their students. I am so appreciative of the GCEE because I know that it's such a powerful organization because so many people, businesses and corporations and individuals invest in there because they know they're investing in my children. They were, I don't want to say just interesting to me, but they gave me so many ideas of how I could teach certain concepts in a way that might really engage my students so that I don't have to do a pure lecturing style because usually with this you put students to sleep but how I can teach in an engaging way that whenever something came up where I felt needs for improvement in my teaching, I signed up for another workshop. When we create any of our workshops we start with looking at the Georgia performance standards. Teachers are required to teach certain concepts throughout K-12 and it makes sense for us to deliver the concepts that teachers are expected to teach. If you come to a fourth grade workshop for instance, the economics we're going to be looking at is specific to the fourth grade GPS standards and I think this helps build teachers comfort level with what they're required to teach in the classroom and it allows them to work on new and creative ways to deliver economic concepts to make it more interesting for their students. We figure a typical middle or high school teacher teaches 200 students a year so you can extrapolate that pretty quickly. 10 years, 2,000 students, it's like compound interest. Once that teacher has been to a workshop, used our materials, year after year that impact is going to be felt in their classroom. The Georgia Council on Economic Education has made a huge difference on me as a teacher. When I first started teaching, I would get behind my podium and talk about something and I had a hard time maybe venturing out and doing a lot of stuff but if it had not been for them, I kind of hate to see what my class would be like because they have had an unbelievable influence on me as a teacher.


Salt is considered a probable cause of stomach cancer, one of the world's leading cancer killers. If their estimate of an 8% increase in risk for every extra gram of salt a day is correct, then in a country like the UK, nearly 1,700 cases of stomach cancer happen every year just because of excess salt intake. And in a country like the US, it would beat thousands more every year. The risk of stomach cancer associated with salt intake appears on par with smoking or heavy alcohol use, but may only be half as bad as opium use or increased total meat consumption based on this study of more than a half million people, which may explain why those eating meatless diets appear to have nearly two-thirds lower risk. We know dietary salt intake is directly associated with the risk of stomach cancer and the higher the intake, the higher the risks, but this meta-analysis went further looking at specific salt-rich foods. Pickled foods, salted fish, processed meat, and miso soup. Habitual consumption of pickled food, salted fish, and processed meat were each associated with about a 25% greater risk of stomach cancer. The pickled foods may explain why Korea appears to have the highest stomach cancer rates in the world, but there was no significant association with the consumption of miso soup. This may be because the carcinogenic effects of the salt are counteracted by the anti-carcinogenic effects of the soy, effectively cancelling out the risk. Even if we made garlicky soup with some scallions thrown in, it may drop our cancer risk even lower. But cancer isn't the primary reason people are told to avoid salt. What about miso soup and high blood pressure? Well, it may be the same kind of thing. The salt in miso is squeezing our blood pressures up, but the soy protein in miso may be relaxing our blood pressures down. So for example, if you compare the effects of soy milk to cow's milk and to make it fairer, compare soy milk to skim milk to avoid the saturated butterfat, soy milk can much more dramatically improve blood pressure among women with hypertension. But would the effect be dramatic enough to counter all the salt in miso? Japanese researchers decided to put it to the test. They followed men and women in their 60s who started out with normal blood pressure and followed them for four years to see who is more likely to be diagnosed with hypertension in that time. Those who had two or more bowls of miso soup a day, or those who had one or less. Two bowls a day would be like adding a half teaspoon of salt to one's daily diet, yet those who ate two bowls or more appeared to have five times the lower risk of becoming hypertensive. So maybe the anti-hypertensive effects of the soy in the miso exceeds the hypertensive effects of the salt.


In this video, we're going to talk about solving exponential equations. This is something we've approached before in this lecture series, for which we use the fact that if you have, like, a to the x is equal to a to the y, that is, if you have these two exponential expressions that are equal to each other, because exponentials are one-to-one, we get that the exponent's x would have to equal y. That's a very common tool you can use to solve exponential equations.  It's not impossible. But if it becomes crazy hard to write one of the sides of the equation as an exponential, like, I mean, how do you write 5 as a power of 2? That's a little bit more challenging. So instead of using this one-to-one approach, the idea here is, well, you could switch it from exponential form to logarithmic form, right? So you have this equation 2x equals 5, you have this base 2, because you want to get x all by itself on the left-hand side. You want to move this exponential base 2 to the other side, so you're going to switch it to its inverse function. So exponential base 2 turns into the logarithm base 2 of 5. And so that would be the answer to this question here, log base 2 of 5, because the log base 2 of 5, what that is computing here, this number is the power of 2 that gives you 5, which is exactly what we need to figure out right here. We need the power of 2, which gives you 5. Now, you probably want to estimate that. You might need to, let's say you need to round it to 3 decimal places or something. It's very likely when you consult your calculator, it doesn't have a log base 2 button or a general log button. So by the change of base form, you can write this as the natural log of 5 divided by the natural log of 2. You can use the common log also if you prefer, but in calculus, you see that the natural log really is the superior log. I mean, there's a reason why we call it natural. The natural log is the naturally occurring logarithm, believe it or not. And as such, I'd say for college hours, we get in the habit of using the natural log right now. You can approximate the natural log of 5 on your calculator, the natural log of 2. While those are both in your memory bank of the calculator, take their quotient. I noticed earlier, I wrote that it was approximate. These things are equal so far. The natural log of 5 divided by the natural log of 2 is equal to log base 2 of 5. But this, when you consult your calculator, you get about 2.322. So those would give you, that is a solution to this one. And that's an appropriate approach to take to this one. I do want to present an alternative approach, right? When you take something like 2x to the 5 right here, another approach that one could take is that, you know, since you had to use the natural log to approximate your calculator, one sort of asks the natural, we have the following thought. What if we just start with taking the natural log of both sides, right? If you take the natural log, you know, as long as you do the same thing to both sides of the equation, what's good for the goose is good for the gander, you're going to get the equation solved, right? Equality will be preserved as long as you go along. And so if we take the natural log of both sides, the nice thing is when you take the natural log of the left-hand side, the third law of logarithms comes into play, right? Exponents inside the logarithm come out as coefficients. And so this becomes x times the natural log of 2 is equal to the natural log of 5. And when you're working with logarithms, an important thing to remember here is that logarithms are just numbers, right? When you look at something like the natural log of 2, that's just a number. A rational number, mind you, but it's just a number. And so if you had something like 2x equals 5, you know exactly what to do. You divide both sides by 2. Well, in this situation, our numbers just have these, you know, there are rational numbers which we're using logarithms to describe them. This is just the natural log of 2 times x is equal to the natural log of 5. It's just a number still, the natural log of 2. So divide both sides by the natural log of 2 so that they cancel on the left-hand side and you get that x equals the natural log of 5 over the natural log of 2, which we saw earlier, kachink, that's the same answer, right? This is just log base 2 of 5. So we got the same answer in a slightly different way, right? And this one seemed like it was a little bit longer process. It's like, well, if you switch to the exponential form you got there immediately, why did you take the natural log of both sides? Well, we're going to see that for more challenging equations, starting off by taking the natural log of both sides is actually a very fruitful strategy. Let's look at a slightly more complicated example. Let's solve the equation 8 equals 3 to the x equals 5. Well, one approach would simply just be to start ripping off everything attached to the x here. So we're going to start off by dividing both sides by 8. So we get 3 to the x equals 5 eighths, like so. And then like we said last time, so what we were trying to do, why do we divide by 8? We're moving the 8 to the other side. We were multiplying by 8. So to move to the other side, you get its inverse operation division by 8. It's the same thing here. We want to move the base 3 to the other side and that's going to give us x equals the log base 3 of 5 over 8, like so. For which, again, if you want to approximate this thing, you can soldier calculator. At the very least, you're going to have to write this probably as the natural log of 5 eighths over the natural log of 3. I should also mention that by the change, by the laws of logarithms number 2 in particular, if you have a fraction inside of a logarithm, that's the same thing as taking a difference of logs. This is the log base 5, excuse me, the natural log of 5 minus the natural log of 8 over the natural log of 3. And so either one of these ones, all three of these things are all the same thing. It doesn't matter which one you use necessarily, but you can soldier calculator. You're going to get negative 0.428. So that's our solution right there. If you want to solve it using this exponential form. But like we saw earlier in the previous example, I claim we can actually solve this very effectively just by taking the natural log of both sides, right? Take the natural log on the left. We take the natural log on the right. And what do we get here? Well, on the left hand side, we have a product of two, two terms 8 times 3 to the x. When you have a natural log of a product, this becomes a sum of natural logs. You get natural log of 8 plus the natural log of 3 to the x equals the natural log of 5. Okay. So then you want to move the natural log of 8 to the other side. It's just a number. So if you're adding the natural log of 8, we'll have to subtract it from both sides. And so we end up with the natural log of 3 to the x is equal to the natural log of 5 minus the natural log of 8. That seems like I've seen that somewhere before, right? Then, like we saw in the previous example, you can use the third law of logarithms to pull out the x. And so you get x times the natural log of 3 is equal to the natural log of 5 minus the natural log of 8. And then you divide both sides by the natural log of 3. And looking there, you get the exact same number. X is equal to the natural log of 5 minus the natural log of 8 divided by the natural log of 3. It's the exact same number. The process is a little bit different. It is. But because of the laws of logarithms or the laws of exponents, whichever you need to use at the time, they're going to be equivalent to each other because if there's two correct ways to solve an equation, guess what? It doesn't matter, right? If they're both right ways, then they'll give you correct answers when you're done. And so that's so you have a little bit of variety on how to solve these things. And like I said, the more complicated the exponential gets, the more this yellow method turns out to be very fruitful. So for example, look at this one right here. Let's solve the equation 5 to the x minus 2 is equal to 3 to the 3x plus 2. In this situation, you'll notice that you have an exponential base 5 on the left-hand side and you have an exponential base 3 on the right-hand side. Why in the world would you ever consider such a thing? Well, first of all, notice if you had exponential equations f of x equals 5x minus 2. So this is an exponential base 5, which has been shifted to the right by two units. If you had g of x, then s3 to the 3x plus 2, right? So this one's an exponential function base 3 where there's been some horizontal compression and shifting and such. And if you want to find the intersection, right, where are these things intersect each other, right? So their graphs, you might see something like the following, like one could look like this, and then the other could look something like this, where there's a point of intersection. Where is that intersection? You'd have to solve the equation f of x equals g of x. So that's the exact situation we'd be looking at something like this. So this is not such an unnatural equation. This is a very natural thing to consider when you are looking at exponential growth and things like that. So how'd you solve this one? Well, my recommendation is that as opposed to the previous case, where you have this mismatch of bases, like the base 5 and base 3, so we should just be like, eh, I don't care. I'm just going to take the natural log of both sides, right? So let's stop trying to compromise on what the base should be. Let's just take it as it is and just take the natural log of both sides. I'm going to take the natural log on the left. I'm going to take the natural log on the right. Because in this situation, it doesn't really matter what the base is, because the natural logs, you can still use logarithms to pull out these exponents. So you're going to get the natural log of 5 to the x minus 2 on the left-hand side by the third law of logarithms. We can pull that exponent out, and we end up with x minus 2 times the natural log of 5, which remember, the natural log of 5 is just a number, right? It's really no different than a 3, right? The next thing, you're going to have the natural log of 3 on the right-hand side raised to the 3x plus 2 power. You can pull that exponent out by the third law, and so you get 3x plus 2 times the natural log of 3. So again, these are just numbers, right? It'd be like if we have a 7 and an 8 or whatever, you can distribute these numbers, okay? For which on the left-hand side, you'll get x times the natural log of 5 minus 2 times the natural log of 5. On the right-hand side, we're going to get 3 times the natural log of 3 times x. That's just the coefficient of x, and then you're going to get a 2 times the natural log of 3. These are just our coefficients right here. We want to combine like terms, so we're going to move this one to the other side, and we're going to move this one to the right-hand side. So we end up with the natural log of 5x minus 3 times the natural log of 3x, okay? On the right-hand side, we get 2 times the natural log of 3 plus 2 times the natural log of 5, all right? And so then on the left-hand side, you'll notice that we have like terms we can add together, or in this case, subtract their coefficients, or another way of thinking about it, you could factor out the common divisor of x right there. So the left-hand side becomes the natural log of 5 minus 3 times the natural log of 3 times x. Let's see, on the right-hand side, I mean, you could factor out the 2 if you want. I mean, it's not super helpful, but you know, when you see a common divisor, I just can't help myself. I have to factor it. And then to get x all by itself, we're going to divide by its coefficient, which is the natural log of 5 minus 3 times the natural log of 3. We do that on the right-hand side as well. In which case, then, we have our solution. x equals 2 times the natural log of 3 plus the natural log of 5 over the natural log of 5 minus 3 times the natural log of 3. There you go, for which you could throw that into a calculator and you would get approximately negative 3.212, which is honestly a perfectly good-looking solution right there. Now, if you had tried to solve this using some different approaches, you might get different things, but I want you to be aware that if you wanted to, you could condense this logarithm, this logarithm expression all into one single logarithm, right? How could you do that? Well, in the top, right, you have a sum of two logarithms, so that would become 2 times the natural log of 3 times 5, which would be 15. In the denominator, right, you have this negative 3 right here. You could bring it inside. So you get the natural log of 5 minus the natural log of 3 cubed, which is 27, okay? Keep on going. You could bring the 2 inside on the top, and so you end up with the natural log of 15 squared, excuse me. And then in the denominator, because you have a difference of logarithms, you could then write that as the natural log of 5 over 27. So let's note here, of course, that 15 squared is 225. So you get the natural log of 225 divided by the natural log of 5 over 27, for which, you know, if you wrote that as a single log, because this is a change of base type formula, your answer turns out to be log base 5 over 27 of 225, right? That's the same number, right? Again, this looks like a little bit confusing here, but when you look at the original equations, like, isn't it obvious? Oh, yeah, you should have been working base, you should have been working log base 527s, right? I mean, you can see kind of a retrospect. It's like, okay, I can see where the 5 came from. I see where the 5 came from, and the 27 was 3 cubed right there. You can kind of see in retrospect, but what I'm trying to say is when you're working with log or with exponential equations, you can kind of just forego any consideration of the bases and just switch over to natural log, because the natural log has all the capacities you need to solve the exponential equations. So don't necessarily worry about the base, just use the natural log. The natural log is your friend, I can promise you that.


What's up everybody? Welcome back to the weekly wrap-up. We are wrapping the week and we are in good mood. My name is Liz Jackson and my name is Ndanya, the gracious host. So we have trending news this week and the first news is YouTuber Nikola Skiyoko Nwaifu wa Boashlidi did an interview in Palituko Extra. Which of my habits makes you angry? He is場amっちゃuchaerel. Hum. So I started in Palitun ataka. Like my wife said, My husband is her husband. Like if my husband is my husband! So you yourself threatened her? Somewhere in Palituku. na wachiaa pwa na tolea sueta na tolea guwa then in his response Nikola skia kwa kasi ma but I have a wife I have a wife Nikia kiki kumbe ya pa unafa una kujuna kujuna toa and netizens were very mad walu kwa na se ma a wife should be a helpa not a house help what do you feel about that can say na help kwa kwa 3 do 3 do say see she's a helper but seriously I feel like I wouldn't want my ma to do that you know you are a grown up I have a grown up wukisha ka socks kitchen so I was a helpa na huku jia pana I'd prefer if it was inge na to ni I hope it was joking se wukuli I hope so too juw it was inge na say right to zita kwa uta choka uta pwa ka I think some habits unazi juwango wukisha ishi namu to there are some people for example I live with you na spenangi bileu wu na kunja socks zako kuna wu nyanazi kunja na namabor wana ka bili like I can express to you ni kwa mbe and then ya beti wacha kunja socks hiwi naka urogi please and then you change plus it also when you're getting into a relationship with someone unim to make up differently you've been away for each other for 2-5 years so kuna witu wam na famo at just but in the same breath do not disrespect your partner just because kutumusie ke mazoyan osana ilia asiatafanya do things the way you do dem wukwi wa pekeako kisha pekeako bili wu na eka nga nga wu kwa laundry basket eka tu kwa laundry basket nga na suali uke kwa nga chavu kwa laundry basket wuna zirusha tu wa muna kunja yaw kwenya pa ya we won't judge you we promise kwa wu kwa wu kwenya kwa wu kwa laundry basket habi to your house habi to your house laundry basket kwa wu kwa wu kwa wu kwa wu anyway mita nipenda mooden ya niko muna siku e enta kwa na time on my hands nga tumgo tu chafu na tu ega e kuna siku zingine bano ume chokato nye 3 e ume tu kwa eventa do wu nimbedo wazu kukunja kwa go chafu on to our next topic tumba man went ahead to post on his instagram that he was physically assaulted by his girlfriend ame kato kumkona mita ru liwa wu kwa wu kwa mtumbaman kabisaen na kuna makuza kisu kuka chest ya ata kandika kwa kwa kasema ame kwa akifiga shamed kusema kwa haji watwa teite kaji bachakaenda kasema an appreciate oga obina oga obina came through for him miyo story aki it's very sad especially kwa most of the time when you talk about gender based violence it's always the man to the woman but also we must not forget that sometimes women are physical towards men same case na tumbilih kwa skwa luko metandikwa na sujuka ame ni bibi ame ni bibi mama one of the two i feel like it is sad and for women our gender based violence towards men sometimes wu ina oba lukiwa una je nanza nga nakituki doko kumrushi ame kono kiji ko manguwo but i do not support gender based violence on both parties i feel like what the girlfriend did is completely wrong and i hope he gets help but you know i am not blaming anyone i am so sorry kwanza tumtumbaman but don't you see the science sometimes you can never tell the science you can be madly in love with someone then people also change in the middle of the relationship but you don't do kumrushi kisundwa kukate siya liyanzana kitu liyanzana kitu naka rushi anisha TV kapa suka what i don't get if someone is physical with you today nva je unafakwa dil breaker fande keep it namos ce manga bach na sites ratunah siya na gori kawanna za kibenti ati nки kaka sobuna edisa wonashanga bult Ngumi asubuwa好a? only two people say this is our only relationship traditions are not what we want This is not the generation of complex relationships wallpaper is violent really really we are lianna it's bad on to the next story it's bad outed来 kwa relationships YouTuber shon aroha who relocated to the UK she was living in kenya then she was in loans wama hibu Kur Please ... Device li semalete Karavan skip backing ... Na Nairofu ya Kaa Sema Mo .... I'm defending my husband and not a random girl Aki wa shiana ... Muiyeh? ... Muiyeh? ... This one I feel to me I feel like Shona Rwa is a good friend ... Muichu ... She is ... You know why ... If Ndanya sayu Kuosinguba anyway ... Ukiyeh Za Bari Kikwa Namtuh If you ever get into a relationship with someone and then you come and tell me how this person is hurting you, see me, I'll be on your side, you're my friend. But you know what, this is where people go wrong, it depends on how you'll come and tell me. Already that is sad news, already that news will hurt me. So, wuki ni breaki ya news, kujuki ni ambiya from a point of we are in this together. Uskujia from a point of unani ataku, ujuju chadi ako, chali ako ni chali akilam tu, then ni don't do that, it's the way you come and tell me the news. Already I'm in love, remember. I'm in love, so come with me, come and team work against this issue, which is cheating, which is not even human. Ni viletu ta package. But I feel the vlogs that Nairofi did after the breakup, na jua ni kama alikwame, start anew. They used to hang out with Shonarwa, takuna video washa ifanya wa ke liya, Nairofi ali wa pranka kuna bolia, ali impranka kuna bolia exwake. They really cried on that video in Shonarwa, alikwa na muriashua. So to me I feel, I feel what went wrong, what we do not know is why their friendship ended in the first place. Yo, we are not really certain, but I also think the same. But I feel kuna... I just feel like Shonarwa is a good friend to be honest. And I did not expect Nairofi to talk like that. Mi rani kwa na expecto tanya mahalza. Amata, I wake me more something. Ya, okay. Kwanza hiyo ya, kwanza kuna ingini ali weka janas. Chuka ma ni medilit? Kuna ingini ali weka jana? Social media influence atana resits. Resits, bana we are pulling up the resits. Kuna ingini ali weka jana was like, muna ungelele anani evo. Ali delete. Ali delete? Ya, she deleted. But according to me I feel, they would really groan each other. Uli kuna watch waki fanya vlogs za Kibera? Wa metua nambali mi, wana ni nuna Nairofi enu wana ni nyu Shonarwa, wali kuna fanya nga vlogs uko Kibera. Wana ingia, wana tafta place ya kushuka ma make up wana fanya ma make up. That's how they blew up. Then waka separate sa zene, Shonarwa started living with a boyfriend in Nigeria yu kukakuana yu o gap. Then waka reunite juzi juzi. So awa ni watu, siya tu ame jwana sayi o ame jwana from Kitambo. Ya. But then ya, if you ever like wrong me I don't think neza kwa dress online. I know what I miss you. I think niya ma zia. Kwanza kama you a relationship. Ya. Nisa awa ali muambia. Mi ya tia ya feel wacha kwingili ya na sasa. Ya wacha kwa chana. If they know friends they stop being in each other's business. Ya. I am not going to say who's team I am in but it's clear. I forgive kwa kwa chana wanguia 12ish. So I forgive. So sa this video that has been trending wacha kwa kwa mazishi. Siya apumbele kofini mewe kwa. And then, you know... They are done for a bit. They are dancing to a bit. Kwa kwa kwa kwa kwa kwa kwa kwa kwa. Iwi 2. Iwi. Alafu. Anitonga. Munya miku fani na kwa na. Marey mu. Marey mu ajafuni kwa. Marey mu. Mama Marey mu alikwa na pera. Alikwa dancer. Ya na. Ya na ni tiktok. Asuza. Kwa kwa kwa kwa. Buni se. Ya. But that was very... I felt that very funny. Kwa kwa kwa kwa mazishi. I feel sometimes there are people who... Instead of crying and mourning, they celebrate the life of that person. Mibu om tuwali wambia perewein. Kiwa idedi, miskujeku liya liya apa. Just, you know, be happy. Kuna watu bede wa nikakwa weel. Just wear a white in my wedding. Be happy, celebrate that.  Don't focus so much on the... I agree but... We do not wanna chungu liya. One at work, one at work, one at chungu liya. It's very awkward. It's very weird. It's very weird and funny at the same time. Kwa kwa kwa kikitu ya kofi, danyelu kuna trend. Awa se wagana waju wana shika kofin, waki dance dance. Na juu na juu awa atu. Wana kwa kwa kwa mewa black. The way they bury people in West Africa, wana heke liya kofin, kwa mabeka, wana dance, wana dance, wana juu yu video nintakwa onyesha. Wee. Neni, watu wana kwa kwa kwa na beliefs, zingi na weird.  I've never thought about it, by the way. I don't know. I don't know. I'm not sure. I don't know. Right. Onto the next story. Hey, watu nini wana bia, guys? Privacy is key. So, there's this Bangalow owner. Bangalow is a a house that's on the ground. So, this man ali jenga nyumbayake then alo vasade nibawake na jenga apartment hana view, hana yua akasema nini wacheni, chen yu kumotoni Bangalow ikoivi ama jenga wukuta hwa inafuni kaji. Inipetiness. You two grown ups talk about it. But who did he talk about it akijenga iya apartment? You do such a thing. Anu niya zaalakiti na kujiwewa. Nini newisha nga ni jenge alafu kujiweke apartment. Ni ni kwa kwa kwa kumotona chunguliwaima naiiba Nini ya talabdani li kwa Nimezo ya kwa mkaku basku, nikiwa nide te ume niweke itselew tiavi umana. Ume niweke like, I think it's wrong but something else wana juki jenga kitu nafa kua probiyo na gawa. Na kwa kwa tha kwen buru na kuza kutapia. There's no legal way wangi subtle issue. I think the only legal way wangi subtle issue is there are some areas that can have restricted ways of construction. Ka ma wastei, kitsuru kwen na tau ufaiku jutu ga apartment. Kwa nga kukutapia, kwa ini gyla na gani but I don't know, nika ma such things happen in juja, nili kwan aso mejuja, We were in Osterjingu before the play school got here, we have some in Osterjingu We had a lot of fun So we went to play in those apartments When we went to the apartments at the canteen We lived in a house that had no apartments Imagine that my house had an apartment One was locked with a door It was locked and there were sleepers Because they were in judah The students in judah used to come here That was very sad It's something I would do If you decide to be Petey So imagine wale ni bazo kuna balkoni sa idea hi wukota Wanna get? Goja Siwi ni balkoni, aini ni bungalo Ya ni wakati, iwi alafsasapa kuna balkoni Bungalo kuna, balko Yeah Bungalo hi wakati Sandram Booth mkotchoa Mike Sunko Relocated to the UK for father studies She posted she'll be there for four years Na kakapu emotional Sunko nolik emotional, post up o Kimusha pae Kimusha al the best, aki mumbi al miss you, my daughter My home will not feel the same again Oh my gosh So, ame acha Dogi za keze langatati kei pa Kama tu? Ame ziwa jana Ni ziwa storyo tu wefkiri umboa Hei, ni poa Oni jo al nasasana The mum now has gone But you know there's Andrei Sandra khan carry her dogs Hei bai de unizal patatana kuna jet Sunko tena mehaya Private jet je umboa My sister work hard Hei, mi na kwa Man is good Siya storyo te tu niliwa na tu we Mi poa kuku kwa namza ziwa kuna do Wezi sketi Sandra atia kuku kwa kuna stress Aja li pa rent tokens zime isha suji ni ni ni Baba ke shakadi, shakadi But I feel Sandra Sandra is very down to us Like I met her for the very first time And she's so, like her energy is so welcome And she's bubbly She's so bubbly I love her so much I really love her And I wish her all the best I wish her all the best I wish her all the best I also wish her all the best Hei, mimi kwa isu storyo te And ma suma umboa ni kakumu kanya She has a dog Nazila magari ma leksas Sandra Siya tanpuperte lea pink Sandra, can you adopt a bouncing baby girl Kulbenya Bouncing baby girls ready for adoption Bouncing baby girls are ready I'm ready to relocate you and start a new life Tupeleka espiambaz UK Espiambaz But all the best Sandra Go Ine chang wajie Wings Wings to fly Wings to fly Wings to fly ni aiku Wings to fly Apana Muntwa qiambua I really have wings and fly I can't wait to see what you have The sky is the limit I want to see it I want to see the space The space is the limit I want to see it Spread your wings and fly Like an eagle Hey, so guys That's it for the weekly recap We're going to wrap up this week kwa mendi, nga kwa jumaan. Kwa mendi, hwa nga tisha ya'ya. Nakudu kilatinu, tu taria chambu wa ptambi yopoENT kwa mendi ya'ya. Nkukumama tiniu ziaxian. Nkukumama tiniu, njahari.