Evaluate 6 log base 8 of 2+( log base 8 of 64)/(3 log base 8 of 4)

$6 \log_{8} (2) + \frac{\log_{8} (64)}{3 \log_{8} (4)}$

Simplify each term.

Logarithm base $8$ of $2$ is $\frac{1}{3}$ .

$6 (\frac{1}{3}) + \frac{\log_{8} (64)}{3 \log_{8} (4)}$

Cancel the common factor of $3$ .

Factor $3$ out of $6$ .

$3 (2) \frac{1}{3} + \frac{\log_{8} (64)}{3 \log_{8} (4)}$

Cancel the common factor.

$3 \cdot 2 \frac{1}{3} + \frac{\log_{8} (64)}{3 \log_{8} (4)}$

Rewrite the expression.

$2 + \frac{\log_{8} (64)}{3 \log_{8} (4)}$

Rewrite $\log_{8} (64)$ as $\log_{8} (2^{6})$ .

$2 + \frac{\log_{8} (2^{6})}{3 \log_{8} (4)}$

Expand $\log_{8} (2^{6})$ by moving $6$ outside the logarithm.

$2 + \frac{6 \log_{8} (2)}{3 \log_{8} (4)}$

Rewrite $\log_{8} (4)$ as $\log_{8} (2^{2})$ .

$2 + \frac{6 \log_{8} (2)}{3 \log_{8} (2^{2})}$

Expand $\log_{8} (2^{2})$ by moving $2$ outside the logarithm.

$2 + \frac{6 \log_{8} (2)}{3 (2 \log_{8} (2))}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6 \log_{8} (2)$ .

$2 + \frac{3 (2 \log_{8} (2))}{3 (2 \log_{8} (2))}$

Cancel the common factors.

Cancel the common factor.

$2 + \frac{3 (2 \log_{8} (2))}{3 (2 \log_{8} (2))}$

Rewrite the expression.

$2 + \frac{2 \log_{8} (2)}{2 \log_{8} (2)}$

Cancel the common factor of $2$ .

Cancel the common factor.

$2 + \frac{2 \log_{8} (2)}{2 \log_{8} (2)}$

Rewrite the expression.

$2 + \frac{\log_{8} (2)}{\log_{8} (2)}$

Cancel the common factor of $\log_{8} (2)$ .

Cancel the common factor.

$2 + \frac{\log_{8} (2)}{\log_{8} (2)}$

Rewrite the expression.

$2 + 1$

Add $2$ and $1$ .

$3$