Natural Logarithm

Definition

The natural logarithm $\ln x$ is the inverse of the exponential function $\exp (x)$. It is the logarithm with base $e$:

$$\ln x={\log }_{e}x.$$

Domain: positive real numbers $x>0$ (occasionally $\ln |x|$ is used to include negative reals). $\ln 0$ is undefined.

Intuition

$\ln x$ measures how many times $e\approx 2.718$ must be raised to a power to reach $x$. Geometrically, $\ln x$ equals the area under the curve $1/t$ from $t=1$ to $t=x$. It converts multiplication into addition, which makes it the natural tool for reasoning about ratios and exponential growth rates.

Formal Description

Standard logarithm rules:

$$\ln (xy)=\ln x+\ln y,\ln \left(\frac{x}{y}\right)=\ln x-\ln y,\ln (x^r)=r\ln x.$$

Special values:

$$\ln 1=0,\ln e=1.$$

Power law via logarithm: For all real $p$,

$$x^p=\exp (p\ln x).$$

Mutual inverses:

$$e^{\ln x}=x(x>0),\ln (e^x)=x.$$

Applications

Used to solve exponential equations (e.g. $e^{rt}=2\Rightarrow t=\frac{\ln 2}{r}$), to define entropy and information content ($-\ln p$), and to linearise power-law relationships in data analysis via log-log plots.

Trade-offs

$\ln x$ is only defined for $x>0$; extending to negative reals or zero requires either $\ln |x|$ (losing injectivity) or complex logarithms (introducing multi-valuedness). Numerically, $\ln x$ is ill-conditioned near $x=0$.

Links

• Growth, Decay and Oscillation — exponential growth/decay