# How do you simplify Log(x+4)=2log(x-2)?

Dec 25, 2015

$x = 5$

#### Explanation:

Simplify the right hand side using the logarithm rule:

$\textcolor{w h i t e}{s s s} a {\log}_{b} \left(c\right) = {\log}_{b} \left({a}^{c}\right)$

$\log \left(x + 4\right) = \log \left({\left(x - 2\right)}^{2}\right)$

Exponentiate both sides, which undoes both logarithms, leaving us with:

$x + 4 = {\left(x - 2\right)}^{2}$

Distribute and simplify.

$x + 4 = {x}^{2} - 4 x + 4$

$0 = {x}^{2} - 5 x$

$0 = x \left(x - 5\right)$

$x = 0 , 5$

However, the answer $x = 0$ is thrown out, since plugging in $0$ in $2 \log \left(x - 2\right)$ would result in having to find the logarithm of a negative number, which is impossible.

Thus, the only valid answer is

$x = 5$

graph{log(x+4)-2log(x-2) [-8.24, 17.07, -3.01, 9.65]}