Properties of Exponential Function

Theorem
Let $$x \in \mathbb{R}$$ 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 $$\mathbb{R}$$ and range $$\left({0 \, . \, . \, \infty}\right)$$.


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

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 range is $$\mathbb{R}$$, 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.