# How do you find the inverse of an exponential function?

Nov 23, 2017

#### Explanation:

Suppose we have: $y = {a}^{x}$

Take logs of both sides

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

But $\ln \left({a}^{x}\right)$ is the same as $x \ln \left(a\right)$ giving:

$\ln \left(y\right) = x \ln \left(a\right)$

Divide both sides by $\ln \left(a\right)$

$\ln \frac{y}{\ln} \left(a\right) = x$

Now swap round the letters $x \mathmr{and} y$ giving:

$\ln \frac{x}{\ln} \left(a\right) = y$

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