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,

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