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 $s$.

$\blacksquare$

Also see

• Bounds of Natural Logarithm

Sources

• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.3 \ (2)$