# How do you condense 2log_3x-3log_3y+log_3 8?

Aug 24, 2016

${\log}_{3} \left(\frac{8 {x}^{2}}{y} ^ 3\right)$

#### Explanation:

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

$\textcolor{\mathmr{and} a n \ge}{\text{Reminder " color(red)(|bar(ul(color(white)(a/a)color(black)(logx+logy=log(xy))color(white)(a/a)|)))" and }}$

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\log x - \log y = \log \left(\frac{x}{y}\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}}$

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

$\Rightarrow 2 {\log}_{3} x = {\log}_{3} {x}^{2} \text{ and } 3 {\log}_{3} y = {\log}_{3} {y}^{3}$

$\Rightarrow {\log}_{3} {x}^{2} + {\log}_{3} 8 = {\log}_{3} \left(8 {x}^{2}\right)$

and ${\log}_{3} \left(8 {x}^{2}\right) - {\log}_{3} {y}^{3} = {\log}_{3} \left(\frac{8 {x}^{2}}{y} ^ 3\right)$