Natural Logarithm

Summary: $\ln (x)$ is the inverse of the exponential function $e^x$, and the rescaling constant that converts any other exponential into base $e$. As an inverse, $\ln (x)$ asks: ”$e$ to the what power gives $x$?”

Key identity: $\ln (xy)=\ln (x)+\ln (y)$: multiplication in the input becomes addition in the output.

Definition

$$\ln (x)=y⟺e^y=x$$

Equivalently, $\ln$ is defined by:

• $e^{\ln (x)}=x$ for $x>0$
• $\ln (e^y)=y$ for all real $y$

It’s only defined for positive $x$, because $e^y>0$ always.

What it actually does

$\ln (x)$ tells you the time it would take a quantity governed by $e^t$ (growing such that velocity = position, starting at 1) to reach the value $x$.

• $\ln (1)=0$ (already there)
• $\ln (2)\approx 0.6931$
• $\ln (e)=1$
• $\ln (3)\approx 1.0986$
• $\ln (8)=\ln (2^3)=3\ln (2)\approx 2.0794$
  • since $\ln (2\cdot 2\cdot 2)=\ln (2)+\ln (2)+\ln (2)=3\ln (2)$

The last point illustrates the headline algebraic identity:

$$\ln (xy)=\ln (x)+\ln (y)\to \ln (x^k)=k\ln (x)$$

This is just the exponential identity $a^{x+y}=a^x{a}^y$ run backwards through the inverse — multiplication in the input becomes addition in the output, which is what makes logarithms useful for compressing dynamic range.

Resolving the mystery constants

In exponential-function we saw every $a^t$ is proportional (but usually not equal) to its own derivative with some base-specific constant $k_a$. That constant turns out to be exactly $\ln (a)$.

The argument:

• any $a^t$ can be rewritten in base $e$.
• The required scaling is whichever $c$ makes $e^c=a$. By definition, $c=\ln (a)$. So:

$$a^t=e^{\ln (a)\cdot t}$$

Differentiating with the chain rule:

$$\frac{d}{dt}{a}^t=e^{\ln (a)t}\cdot \ln (a)=a^t\cdot \ln (a)$$

So the “mystery constant” is no mystery — it’s the natural logarithm of the base.

Derivative

By inverse function differentiation:

$$\frac{d}{dx}\ln (x)=\frac{1}{x}$$

This is striking: integrating $1/x$ gives $\ln (x)$, which is why the natural log shows up in integrals of rational functions and in the harmonic series asymptotics.

Why “natural”

It’s natural in the sense that it’s the logarithm whose derivative is just $1/x$ with no extra constant. Any other base ${\log }_a(x)=\ln (x)/\ln (a)$, so picking base $e$ removes the multiplicative clutter — the same reason calculus prefers $e^t$ over $a^t$.

Sources

• What’s so special about Euler’s number e?