# How do you find the inverse of y = (log_2 x) +2?

Jun 21, 2016

This process reflects the equation about $y = x$

y=10^(log_10(2)(x-2)

#### Explanation:

Using: ${\log}_{2} \left(x\right) \to {\log}_{10} \frac{x}{\log} _ 10 \left(2\right)$

$\text{ } y = {\log}_{10} \frac{x}{\log} _ 10 \left(2\right) + 2$

$\textcolor{b r o w n}{\text{Subtract 2 from both sides}}$

$\text{ } y - 2 = {\log}_{10} \frac{x}{\log} _ 10 \left(2\right)$

$\textcolor{b r o w n}{\text{Multiply both sides by } {\log}_{10} \left(2\right)}$
$\textcolor{w h i t e}{\frac{2}{2}}$

$\text{ } {\log}_{10} \left(2\right) \left(y - 2\right) = {\log}_{10} \left(x\right)$
" "color(brown)(|ul(" ")|)
$\text{ } \textcolor{b r o w n}{\downarrow}$
$\textcolor{b r o w n}{\text{ Let "log_10(2)(y-2)" be } z}$

${\log}_{10} \left(x\right) = z \text{ "vec("another way of writing this is")" } x = {10}^{z}$
$\textcolor{w h i t e}{\frac{2}{2}}$
$\textcolor{b r o w n}{\text{So by substitution for "z" we have:}}$
$\textcolor{w h i t e}{\frac{2}{2}}$
" "=>x=10^(log_10(2)(y-2)

$\textcolor{b r o w n}{\text{Swap the letters "x" and "y" round and we have:}}$
$\textcolor{w h i t e}{\frac{2}{2}}$

" "color(blue)(y=10^(log_10(2)(x-2))
$\textcolor{w h i t e}{\frac{2}{2}}$