# How do you evaluate log_27 9?

Aug 7, 2016

$x = \frac{2}{3}$

#### Explanation:

In log form the question being asked is ...

"To what power must 27 be raised to give 9?#

You should recognise 27 as ${3}^{3}$ and 9 as ${3}^{2}$

$3 = \sqrt[3]{27} = {27}^{\frac{1}{3}}$

$9 = {3}^{2} = {\left(\sqrt[3]{27}\right)}^{2} = {27}^{{\left(\frac{1}{3}\right)}^{2}} = {27}^{\frac{2}{3}}$

OR

${\log}_{27} 9 = x \Rightarrow \text{ } {27}^{x} = 9$

${3}^{3 x} = {3}^{2}$

Equating exponents: $3 x = 2$

$x = \frac{2}{3}$