# How do you evaluate log_15(15)?

Dec 3, 2015

${\log}_{a} \left(a\right) = 1$ holds for any positive integer $a$.

#### Explanation:

You are basically searching for $x$ so that

${\log}_{15} \left(15\right) = x$

The inverse function of the ${\log}_{15} \left(z\right)$ is ${15}^{z}$which means that
${\log}_{15} \left({15}^{z}\right) = z$ and also ${15}^{{\log}_{15} \left(z\right)} = z$ always hold.

This means that to "get rid" of the logarithmic term, you can perform ${15}^{z}$ on both sides of your equation:

$\textcolor{w h i t e}{\times x} {15}^{{\log}_{15} \left(15\right)} = {15}^{x}$

$\iff \textcolor{w h i t e}{\times \times x} 15 = {15}^{x}$

$\iff \textcolor{w h i t e}{\times \times x} {15}^{1} = {15}^{x}$

So you can see that $x = 1$ which means that

${\log}_{15} \left(15\right) = 1$