# How do you solve the equation 1/3logx=log8?

Dec 3, 2016

$x = 512$

#### Explanation:

Using the $\textcolor{b l u e}{\text{laws of logarithms}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\log {x}^{n} \Leftrightarrow n \log x} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\Rightarrow \frac{1}{3} \log x = \log {x}^{\frac{1}{3}}$

$\Rightarrow \log {x}^{\frac{1}{3}} = \log 8$

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder }} \textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{\log x = \log y \Rightarrow x = y} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\Rightarrow {x}^{\frac{1}{3}} = 8$

$\textcolor{b l u e}{\text{cubing both sides}}$

$\Rightarrow x = {8}^{3} = 512$