# What is the inverse of y= log_(1/2) (x+4) ?

Mar 4, 2018

The inverse is $y = {\left(\frac{1}{2}\right)}^{x} - 4$

#### Explanation:

To find the inverse, switch $x$ with $y$ and vice versa, then solve for $y$. To convert out of $\log$ form, make it exponential form.

$\textcolor{w h i t e}{\implies} y = {\log}_{\frac{1}{2}} \left(x + 4\right)$

$\implies \textcolor{red}{x} = {\log}_{\textcolor{b l u e}{\frac{1}{2}}} \textcolor{g r e e n}{\left(y + 4\right)}$

$\textcolor{w h i t e}{\implies} \textcolor{g r e e n}{y + 4} = {\textcolor{b l u e}{\left(\frac{1}{2}\right)}}^{\textcolor{red}{x}}$

$\textcolor{w h i t e}{\implies} y = {\left(\frac{1}{2}\right)}^{x} - 4$

Here's a diagram of the graphs (I included the line $y = x$ to show the reflection):