Natural logarithm
= Natural logarithm
{title2=$\ln$}
{wiki}
The <natural logarithm> $\ln x$ is the <logarithm> with base <Euler's number>, and the <inverse function> of the real <exponential function>. A particularly geometric definition is
$$
\ln x=\int_1^x\frac{dt}{t},\qquad x>0.
$$
For $x>1$, it is the <area> under the <hyperbola> $y=1/t$ from $1$ to $x$. For $0<x<1$, the reversed <integration> limits give a negative value. The <fundamental theorem of calculus> gives $(\ln x)'=1/x$, so the <natural logarithm> is increasing, while $(\ln x)''=-1/x^2$ gives <concavity>.
\Image[/exponential-logarithm-geometry.png]
{title=<Exponential function> and <natural logarithm> graphs reflect across $y=x$; equal multiplicative intervals have equal logarithmic areas}
The <change of variables> $t=au$ shows $\int_a^{ab}dt/t=\int_1^bdu/u=\ln b$ for $a,b>0$. Thus the intervals from $1$ to $2$ and from $2$ to $4$ each contribute <area> $\ln2$, explaining $\ln4=2\ln2$. Equal ratios give equal increments on a <logarithmic scale>, even though the ordinary differences grow. This makes the <natural logarithm> useful for comparing multiplicative changes and solving for time in <exponential growth> or <exponential decay>. The <integral> construction and <inverse function> properties are discussed in https://openstax.org/books/calculus-volume-1/pages/6-7-integrals-exponential-functions-and-logarithms[OpenStax, the natural logarithm as an integral].