# Exponential is Strictly Convex

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## Theorem

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

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

Then:

- The function $f \left({x}\right) = \exp x$ is strictly convex.

## Proof

By definition, the exponential function is the inverse of the natural logarithm function.

From Logarithm is Strictly Increasing, $\ln x$ is strictly increasing.

From Logarithm is Strictly Concave, $\ln x$ is strictly concave.

The result follows from Inverse of Strictly Increasing Strictly Concave Real Function is Strictly Convex.

$\blacksquare$