# How to find the inverse function of h(x)= 2^x ?

Apr 10, 2018

$h \left(x\right) = {\log}_{2} \left(x\right)$

#### Explanation:

The find the inverse of an invertible function $y = f \left(x\right)$, swap $y$ and $x$ and solve for $y$.

We have $h \left(x\right) = y = {2}^{x}$. Swapping $x$ and $y$ gives $x = {2}^{y}$. We now wish to solve for $y$.

Take the log with base 2 of both sides of the equation to free the $y$ variable.

$x = {2}^{y}$
${\log}_{2} \left(x\right) = {\log}_{2} \left({2}^{y}\right) = y$

Thus, $y = h \left(x\right) = {\log}_{2} \left(x\right)$. See that we used the fact that ${\log}_{n} \left({n}^{x}\right) = x$.