## 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 $x$?