Exponential

Exponential $\exp$ is solution of equation

$\exp'(x)=\exp(x)$,

$\exp(0)=1$

Where "prime" demotes the derivative.

Notation $\exp(x)=e^x$

is also used; constant

$ \displaystyle \mathrm e=\sum_{n=0}^\infty \frac{1}{n!} \approx 2.71828182846 $

The same function is called also "natural exponent" or "exponential to base $\mathrm e$, in order to distinguish it from exponential to other base $b$, denoted as

$\exp_b(z)=b^z=\exp\big(\ln(b) z\big)$

where $\ln$ denotes the natural logarithm,

$\ln=\exp^{-1}$

where the superscript at the name of function indicates its iterate; logarithm is minus first iterate of the exponent, id set, the increase function. In wide range of values of $z$, the identity holds,

$\exp\big(\ln(z)\bit)=z$