= Exponential function
{title2=$e^x$}
{wiki}
The real <exponential function> $\exp(x)=e^x$ converts addition into multiplication:
$$
\exp(x+y)=\exp(x)\exp(y),\qquad \exp(0)=1,\qquad \frac{d}{dx}\exp(x)=\exp(x).
$$
Its values are positive, and it defines a <bijection> from the <real numbers> to the positive <real numbers>. The <natural logarithm> is its <inverse function>. These properties connect arithmetic with <calculus>: adding an amount to the input multiplies the output by a fixed factor, and the local rate of change equals the current value. The <exponential series> gives one construction; an equivalent characterization is the unique <solution of a differential equation> $y'=y$, $y(0)=1$.
For a concrete example, $y(t)=10e^{0.2t}$ starts at $10$ and satisfies $y'=0.2y$. The absolute <derivative> increases with $y$, while the relative rate $y'/y$ stays $0.2$. At $t=5$, $y=10e\approx27.1828$ and $y'\approx5.43656$. The <exponential function> therefore models accumulation proportional to what is already present, rather than addition of a fixed amount at every step. A positive constant rate gives <exponential growth>; a negative one gives <exponential decay>.
Every positive real base $b\ne1$ fits the same construction through $b^x=e^{x\ln b}$. Thus its <derivative> is $(\ln b)b^x$: base $e$ is special because $\ln e=1$. The <exponential function> is also unchanged in form by <differentiation> and <integration>, helping explain its repeated appearance in <linear ordinary differential equations>. Its real definition, inverse relationship, and derivative rules are developed in https://openstax.org/books/calculus-volume-1/pages/6-7-integrals-exponential-functions-and-logarithms[OpenStax, integrals, exponential functions, and logarithms].
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