# If log_b x = 2/3log_b 27 + 2 log_b 2 -log_b 3, what is x?

Sep 19, 2016

$x = \textcolor{g r e e n}{12}$

#### Explanation:

Things you need to know and remember:
$\textcolor{w h i t e}{\text{XXX}} {\log}_{b} \left({p}^{q}\right) = q \cdot {\log}_{b} p$
and
$\textcolor{w h i t e}{\text{XXX}} {\log}_{b} \left(p\right) + {\log}_{b} \left(q\right) = {\log}_{b} \left(p q\right)$

$\textcolor{red}{\frac{2}{3} {\log}_{b} 27}$
$\textcolor{w h i t e}{\text{XXX}} = \frac{2}{3} {\log}_{b} {3}^{3} = \frac{2}{3} \cdot 3 {\log}_{b} 3 = \textcolor{red}{2 {\log}_{b} 3}$

$\textcolor{b l u e}{2 {\log}_{b} 2}$
$\textcolor{w h i t e}{\text{XXX}} = {\log}_{b} {2}^{2} = \textcolor{b l u e}{{\log}_{b} 4}$

$\textcolor{red}{\frac{2}{3} {\log}_{b} 27} + \textcolor{b l u e}{2 {\log}_{b} 4} - \textcolor{b l a c k}{{\log}_{b} 3}$
$\textcolor{w h i t e}{\text{XXX}} = \textcolor{red}{2 {\log}_{b} 3} + \textcolor{b l u e}{{\log}_{b} 4} - \textcolor{b l a c k}{{\log}_{b} 3}$

$\textcolor{w h i t e}{\text{XXX}} = {\log}_{b} 3 + {\log}_{b} 4$

$\textcolor{w h i t e}{\text{XXX}} = {\log}_{b} 12$

Therefore
$\textcolor{w h i t e}{\text{XXX}} {\log}_{b} x = \frac{2}{3} {\log}_{b} 27 + 2 {\log}_{b} 2 - {\log}_{b} 3$

$\textcolor{w h i t e}{\text{XXX}} \Rightarrow {\log}_{b} x = {\log}_{b} 12$

$\textcolor{w h i t e}{\text{XXX}} \Rightarrow x = 12$