## Natural Logarithm revisited

So recall we introduced the natural logarithm. Integrals interpret the natural logarithm as $\int t^{-1}\,\mathrm{d}t$, and we will explore that in this post.

1. The natural logarithm satisfies
(1)$\displaystyle\frac{\mathrm{d}\ln(x)}{\mathrm{d}x}=\frac{1}{x}$
By the fundamental theorem of calculus, we can consider a different definition
(2)$\displaystyle\ln(x)=\int^{x}_{1}\frac{\mathrm{d}t}{t}$
Observe that, by (1), the first derivative is positive for $x\geq0$. Also by differentiating it again, we get
(3)$\displaystyle\frac{\mathrm{d}^{2}\ln(x)}{\mathrm{d}x^{2}}=\frac{-1}{x^{2}}$
Thus $\ln(x)$ is concave down but increasing. Thus its sketch should look like:

So What?!? Well, we have an effective method to calculate the natural logarithm of any number! As a matter of fact, we can show a doodle what the logarithm corresponds to:

The curve $f(x)=1/x$ is drawn is blue, and the area between $x=1$ and $x=4$ is shaded in light blue. That light blue region’s area corresponds to the natural logarithm.

WARNING: The blue curve doodled is the derivative of the natural logarithm. This is a subtle point which some students miss, and believe the blue curve is the natural logarithm. This tempting mistake must be avoided!

2. What about $\mathrm{e}$? Recall that $\mathrm{e}$ is the number satisfying
(4)$\displaystyle\ln(\mathrm{e})=1$
The integral form of this statement is
(4′)$\displaystyle\int^{\mathrm{e}}_{1}\frac{\mathrm{d}t}{t}=1$
We can doodle this situation as

We want the shaded green region to have an area equal to 1.