Natural Logarithm

1. So we introduced the Exponential function and considered the notion of inverse functions and their derivatives. But what is the inverse function for y=\exp(x)?

Tradition dictates the inverse function for \exp(x) be written \ln(y) and be called the “Natural Logarithm”.

Let y=\exp(x), we find the derivative to \ln(y) to be

(1)\displaystyle\frac{\mathrm{d}\ln(y)}{\mathrm{d}y}=\frac{1}{\exp(x)}=\frac{1}{y}

We also see \exp(0)=1 implies \ln(1)=0. Similarly \exp(1)=\mathrm{e}\approx 2.7, we find \ln(e)=1.

Similarly, since \exp(a+b)=\exp(a)\cdot\exp(b) we find \ln(ab)=\ln(a)+\ln(b).

The remaining exponential identity \exp(-a)=1/\exp(a) gives us \ln(a^{-1})=-\ln(a).

From these statements, we may deduce everything about \ln(-).

Remark. In practice, we typically consider \displaystyle \ln\left(\frac{x-1}{x+1}\right) when performing calculations. The Continued Fraction form [wikipedia.org] gives a quick calculation when combined with the logarithmic identities.

2. Limit Definition. Recall we had the exponential map defined as a limit as
(2)\displaystyle\exp(x)=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}
If we let
(3)\displaystyle y_{n}:=\left(1+\frac{x}{n}\right)^{n}
then we observe that
(4a)\displaystyle y_{n}^{1/n}=\left(1+\frac{x}{n}\right)
subtracting 1 from both sides and multiplying through by n yields
(4b)\displaystyle n(y_{n}^{1/n}-1)=x
Taking n\to\infty we get
(4c)\displaystyle \lim_{n\to\infty}n(y^{1/n}-1)=x=\ln(y)
However, we are fudging a little, but the general idea is correct.

3. Tricky Derivative. Consider the function
(5)\displaystyle g(x)=x^{x}
What is its derivative? We can rewrite it as
(5)\displaystyle g(x)=\exp\left(\ln\left(x^{x}\right)\right)=\exp(x\ln(x))
How to find this derivative? We use the chain rule!

Thus
(6)\displaystyle g'(x)=\exp(x\ln(x))\cdot\frac{\mathrm{d}(x\ln(x))}{\mathrm{d}x}
After some computation we find
(7)\displaystyle g'(x)=\exp(x\ln(x))\cdot(1+\ln(x))=(1+\ln(x))x^{x}
We couldn’t have done it without the natural logarithm!

Exercise 1. What’s the derivative of x^{\cos(x)} with respect to x?

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About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.
This entry was posted in Exponential Map, Inverse Function Theorem, Natural Logarithm. Bookmark the permalink.

3 Responses to Natural Logarithm

  1. Pingback: Natural Logarithm revisited | My Math Blog

  2. Pingback: [Index] Calculus of a Single Variable | My Math Blog

  3. Pingback: [Index] Calculus in a Single Variable | My Math Blog

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