# Upper Bound of Natural Logarithm/Corollary

## Theorem

Let $\ln y$ be the natural logarithm of $y$ where $y \in \R_{>0}$.

Then:

$\forall s \in \R_{>0}: \ln x \le \dfrac {x^s} s$

## Proof

 $\ds s \ln x$ $=$ $\ds \ln {x^s}$ Logarithm of Power $\ds$ $\le$ $\ds x^s - 1$ Upper Bound of Natural Logarithm $\ds$ $\le$ $\ds x^s$

The result follows by dividing both sides by $s$.

$\blacksquare$