Properties of Exponential Function

Theorem
Let $x \in \R$ be a real number.

Let $\exp x$ be the exponential of $x$.

Then:
 * $\exp 0 = 1$ and $\exp 1 = e$, where $e$ is Euler's number, i.e. $2.718281828\ldots$;
 * The function $f \left({x}\right) = \exp x$ is strictly increasing and convex;
 * $\exp x \to +\infty$ as $x \to +\infty$ and $\exp x \to 0$ as $x \to -\infty$.

Thus the exponential function has domain $\R$ and image $\left({0 \, . \, . \, \infty}\right)$.


 * $\forall x > 0: \exp \left({\ln x}\right) = x$ and $\forall x \in \R: \ln \left({\exp x}\right) = x$.

Proof

 * $\exp 0 = 1$ and $\exp 1 = e$:

These follow directly from the fact that the exponential function is the inverse of the natural logarithm function:


 * 1) $\ln 1 = 0$;
 * 2) $\ln e = 1$ from the definition of Euler's number.


 * $\exp x$ is strictly increasing and convex:

This follows directly from Inverse of Convex Strictly Monotone Function and the fact that $\ln x$ is strictly increasing and concave.


 * $\exp x \to +\infty$ as $x \to +\infty$ and $\exp x \to 0$ as $x \to -\infty$:

This follows from the definition of an inverse mapping.

The domain of $\ln x$ is domain $\left({0 \, . \, . \, \infty}\right)$ and its image is $\R$, hence the result.


 * $\exp \left({\ln x}\right) = x$ and $\ln \left({\exp x}\right) = x$:

These follow directly from the fact that the exponential function is the inverse of the natural logarithm function.