How do you evaluate log_216 6?

Aug 29, 2016

$\frac{1}{3}$

Explanation:

${6}^{2} = 36$

${6}^{3} = 36 \cdot 6 = 180 + 36 = 216$

So ${216}^{\frac{1}{3}} = 6$

and ${\log}_{216} 6 = \frac{1}{3}$

Aug 29, 2016

${\log}_{216} 6 = \frac{1}{3}$

Explanation:

By the $\log$ rule ${\log}_{a} \left(n\right) = \frac{\log n}{\log a}$:

${\log}_{216} \left(6\right) = \frac{\log 6}{\log 216} = \frac{\log \left({6}^{1}\right)}{\log \left({6}^{3}\right)}$

Now use the rule $\log \left({a}^{n}\right) = n \log a$ and simplify:

$= \frac{1 \log 6}{3 \log 6}$

$= \frac{1}{3}$