23.5 Natural log: ln

In the first part of the chapter, we will get to be introduced to logarithms. We will learn that Logarithms like log2(16)=4\log_{2}{(16)}=4 simply mean that the base 2 is multiplied 4 times to get the number inside the parenthesis which is 16. To simply put, logarithms are the opposite of exponents, because instead of finding the resulting number from raising your base to an exponent, you are looking for the exponent that is used to raise the base to get a particular number.

Then in the second part of the chapter, we will learn how to convert an expression in logarithmic form like log6(216)=3\log_{6}{(216)}=3 or log2(16)=4\log_{2}{(16)}=4, into their exponential form, 63=2166^3= 216 and 24=162^4 = 16. We will have plenty of examples to do just that.

In the next part we will learn how to evaluate logarithmic expression without the aid of a calculator. To do that, we will be introduced to the common logarithms and natural log. The common logarithms are logs that have the base 10 like log (100) = 2. Natural log on one hand is a log that has the base ee like Ine(5)In_{e} (5).

In the sixth part of the chapter we will be introduced the change base formula .This formula will be used to transform one logarithmic expression in to another expression but with a different base.

In the proceeding parts we will learn how to convert an expression in its exponential form into the logarithmic form. Then we will learn how to apply logarithms in solving equations. From here we will be introduced to logarithmic equations and logarithmic functions. We will learn that the domain and range of the logarithmic functions is the positive numbers and the real numbers respectively.

Last but not the least, we will also be introduced to one of the rules of logarithm, the product rule which simply states that logbAC = logbA + logbC. We will get to have more examples that will illustrate this rule more elaborately.

Natural log: ln


• Definition of `` natural logarithm"" and mathematical constant ``e":":

Recall: common logarithms = log with base 10"``10"
example:log3=log103 \log3 = \log_{10}3
natural logarithms = log with base ``e""
example:ln5=loge5 \ln5 = \log_e5

Like π"``\pi" , a mathematical constant equal to 3.14….., `` e"" is just another mathematical constant equal to 2.71…. .

Significance of π"``\pi" : we use it in circle calculations:
example: areacircle=πr2 area_{circle} = \pi r^2
circumferencecircle=2πr circumference_{circle} = 2 \pi r
Significance of ``e"" : we use it mostly in calculus. `` e"" is a unique number such that the slope of tangent line at every point on the graph of f(x)=ex f(x) = e^x is equal to the y-value of the point.

    • a)
      by using the LOG key on a calculator.
    • b)
      by using the LN key on a calculator.
  • 2.
    Without using a calculator, evaluate:
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Natural log: ln

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