# How do you calculate log_(343) 49?

May 4, 2016

${\log}_{343} 49 = \frac{2}{3}$ (method below)

#### Explanation:

Notice that
$\textcolor{w h i t e}{\text{XXX}} 49 = {7}^{2}$
and
$\textcolor{w h i t e}{\text{XXX}} 343 = {7}^{3}$

and remember that ${\log}_{b} a = k$ is equivalent to ${b}^{k} = a$

Let $\textcolor{b l u e}{{\log}_{343} 49 = k}$

$\textcolor{w h i t e}{\text{XXX}} \Rightarrow {343}^{k} = 49$

$\textcolor{w h i t e}{\text{XXX}} \Rightarrow {\left({7}^{3}\right)}^{k} = {7}^{2}$

$\textcolor{w h i t e}{\text{XXX}} \Rightarrow {7}^{3 k} = {7}^{2}$

$\textcolor{w h i t e}{\text{XXX}} \Rightarrow 3 k = 2$

$\textcolor{w h i t e}{\text{XXX}} \Rightarrow \textcolor{b l u e}{k = \frac{2}{3}}$