# What is the inverse function of f(x) =10^x?

Oct 23, 2015

$y = {\log}_{10} \left(x\right)$

#### Explanation:

Let $\text{ } y = {10}^{x}$

Take logs of both sides giving:

$\ln \left(y\right) = \ln \left({10}^{x}\right)$

Or better still use log to base 10 giving:

${\log}_{10} \left(y\right) = {\log}_{10} \left({10}^{x}\right)$

But ${\log}_{10} \left({10}^{x}\right) = x \times {\log}_{10} \left(10\right)$

and ${\log}_{10} \left(10\right) = 1$

Giving:

$\log \left(y\right) = x$

Now swap $x$ and $y$ around giving

$y = {\log}_{10} \left(x\right)$