Evaluate log base 4 of 81

$\log_{4} (81)$

Rewrite $\log_{4} (81)$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (81)}{\log (4)}$

Rewrite $\log (81)$ as $\log (3^{4})$ .

$\frac{\log (3^{4})}{\log (4)}$

Expand $\log (3^{4})$ by moving $4$ outside the logarithm.

$\frac{4 \log (3)}{\log (4)}$

Rewrite $\log (4)$ as $\log (2^{2})$ .

$\frac{4 \log (3)}{\log (2^{2})}$

Expand $\log (2^{2})$ by moving $2$ outside the logarithm.

$\frac{4 \log (3)}{2 \log (2)}$

Cancel the common factor of $4$ and $2$ .

Factor $2$ out of $4 \log (3)$ .

$\frac{2 (2 \log (3))}{2 \log (2)}$

Cancel the common factors.

Factor $2$ out of $2 \log (2)$ .

$\frac{2 (2 \log (3))}{2 (\log (2))}$

Cancel the common factor.

$\frac{2 (2 \log (3))}{2 \log (2)}$

Rewrite the expression.

$\frac{2 \log (3)}{\log (2)}$

Simplify $2 \log (3)$ by moving $2$ inside the logarithm.

$\frac{\log (3^{2})}{\log (2)}$

Raise $3$ to the power of $2$ .

$\frac{\log (9)}{\log (2)}$

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (9)}{\log (2)}$

Decimal Form:

$3.16992500 \dots$