# How do you solve log_9 4+2 log_9 5=log_9 w?

Mar 21, 2016

$w = 100$

#### Explanation:

$1$. Use the log property, ${\log}_{\textcolor{p u r p \le}{b}} \left({\textcolor{red}{m}}^{\textcolor{b l u e}{n}}\right) = \textcolor{b l u e}{n} \cdot {\log}_{\textcolor{p u r p \le}{b}} \left(\textcolor{red}{m}\right)$, to rewrite $2 {\log}_{9} 5$.

${\log}_{9} 4 + 2 {\log}_{9} 5 = {\log}_{9} w$

${\log}_{9} 4 + {\log}_{9} {5}^{2} = {\log}_{9} w$

$2$. Use the log property, ${\log}_{\textcolor{p u r p \le}{b}} \left(\textcolor{red}{m} \cdot \textcolor{b l u e}{n}\right) = {\log}_{\textcolor{p u r p \le}{b}} \left(\textcolor{red}{m}\right) + {\log}_{\textcolor{p u r p \le}{b}} \left(\textcolor{b l u e}{n}\right)$ to simplify the left side of the equation.

${\log}_{9} \left(4 \cdot {5}^{2}\right) = {\log}_{9} w$

${\log}_{9} \left(100\right) = {\log}_{9} w$

$3$. Since the equation now follows a "$\log = \log$" situation, where the bases are the same on both sides, rewrite the equation without the "log" portion.

$100 = w$

$\textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} w = 100 \textcolor{w h i t e}{\frac{a}{a}} |}}}$