# How do you find \log _ { 6} ( x + 4) = \log _ { 6} 7?

Mar 2, 2018

The answer is $x = 3$.

#### Explanation:

Use the definition of a logarithm:

$\textcolor{w h i t e}{\implies} {\log}_{\textcolor{red}{a}} \left(\textcolor{g r e e n}{b}\right) = \textcolor{b l u e}{x} q \quad \iff q \quad {\textcolor{red}{a}}^{\textcolor{b l u e}{x}} = \textcolor{g r e e n}{b}$

In this case, $a$ is $6$, $b$ is $x + 4$, and $x$ is ${\log}_{6} \left(7\right)$. It sounds kind of confusing, but I'll use colors so it's easier to see:

$\textcolor{w h i t e}{\implies} {\log}_{\textcolor{red}{6}} \left(\textcolor{g r e e n}{x + 4}\right) = \textcolor{b l u e}{{\log}_{6} \left(7\right)}$

$\implies {\textcolor{red}{6}}^{\textcolor{b l u e}{{\log}_{6} \left(7\right)}} = \textcolor{g r e e n}{x + 4}$

Now, the $6$ and ${\log}_{6}$ cancel out:

$\textcolor{w h i t e}{\implies} {\textcolor{red}{\cancel{\textcolor{b l a c k}{6}}}}^{\textcolor{red}{\cancel{\textcolor{b l a c k}{{\log}_{6}}}} \left(7\right)} = x + 4$

$\implies 7 = x + 4$

$\textcolor{w h i t e}{\implies} 3 = x$

The answer is $x = 3$.