# How do you evaluate log_81 3?

Nov 2, 2016

${\log}_{81} 3 = \frac{1}{4}$

#### Explanation:

${\log}_{81} 3 = x$

Rewrite as an exponential. Remember, the answer to a log is the exponent. In this case $x$ is the exponent, and $81$ is the base.

${81}^{x} = 3$

Find a common base for both sides, which is $3$.
$81 = {3}^{4}$, so by substitution...

${\left({3}^{4}\right)}^{x} = 3$

Use the exponent rule ${\left({x}^{a}\right)}^{b} = {x}^{a b}$

${3}^{4 x} = {3}^{1}$

$4 x = 1$

$x = \frac{1}{4}$

Nov 2, 2016

${\log}_{81} \left(3\right) = \frac{1}{4}$

#### Explanation:

Since $81$ is much larger than $3$, our answer will be a decimal, so let's think of this problem in the opposite sense: ${\log}_{3} \left(81\right)$. ${3}^{4} = 81$, so ${\log}_{3} \left(81\right) = 4$.

Using the law of exponents, we know that if ${a}^{m} = n$, then ${n}^{\frac{1}{m}} = a$. So using this rule, we know that ${81}^{\frac{1}{4}} = 3$, so ${\log}_{81} \left(3\right) = \frac{1}{4}$