The logarithm

The Euler e-function sends every real number to a positive height. Pick one such height. Which real number was sent there? A logarithm already reads a general positive base backwards. Here the base stays $e$, so a product of heights has to be read from the addition formula.

Sit the pair across the diagonal

This picture keeps the input on the real line. The curve passes through height $1$, rises to the right, falls toward the axis on the left, and never crosses it. The extra cut is the one the chosen height must satisfy: it has to be positive. Drag the input. One point rides the exponential. The other is that point reflected across the diagonal.

On the reals the Euler e-function is continuous and strictly increasing, so for every $y>0$ the equation

$y=e^x$

has exactly one real solution. That solution is written

$x=\ln y,$

and it is called the natural logarithm. Swapping the letters gives

$y=\ln x.$

It is the inverse of $y=e^x$. Reflecting that graph across the diagonal produces the graph of $y=\ln x$.

A product of heights becomes a sum

The addition formula says that a sum in the exponent is a product of heights,

$e^{u+v}=e^u{e}^v.$

For positive real numbers the inverse reads that identity backwards. This board freezes both numbers positive, so each one has a logarithm. Drag them. The mark at the product has height equal to the sum of the two heights, and that mark lies on the curve.

For all positive real numbers $x$ and $y$,

$\ln (xy)=\ln x+\ln y.$

The other logarithm laws belong with a general positive base.

The drop near zero is slow

To the right, the height of $y=\ln x$ grows without a bound. Toward zero from above, the height falls without a bound:

${\lim }_{x\to +0}\ln x=-\infty ,{\lim }_{x\to +\infty }\ln x=+\infty .$

The fall looks steeper than it is, until a positive power is allowed to weigh it. Drag the input toward zero, and drag that power. It stays positive. The point on the logarithm drops. The other point stays near the axis. At input $1$ the two points meet.

For every real number $\alpha >0$,

${\lim }_{x\to +0}{x}^{\alpha }\ln x=0.$

That is why $y=\ln x$ approaches negative infinity extremely slowly near $x=0$.

One step rises by the reciprocal

For a positive input the slope is no longer the height itself. Drag the input. It stays positive. The horizontal step has length $1$. The vertical step equals the reciprocal of the input. At input $1$ the height is $0$, and the vertical step has length $1$.

For every real $x>0$,

$\frac{\mathrm{d}\ln x}{\mathrm{d}x}=\frac{1}{x}.$

The natural logarithm answers the opening question for the Euler e-function: each positive height comes from exactly one real input. The reflection builds the graph, a product of positive numbers becomes a sum, the fall toward zero is slower than any positive power, and on the positive reals the slope is the reciprocal of the input. A positive base other than $e$ asks the same questions of a general exponential function.