Precalculus Examples

$\frac{\log_{4} (64)}{\log_{16} (4)}$

Step 1

Rewrite $\log_{4} (64)$ as $\log_{4} (2^{6})$ .

$\frac{\log_{4} (2^{6})}{\log_{16} (4)}$

Step 2

Expand $\log_{4} (2^{6})$ by moving $6$ outside the logarithm.

$\frac{6 \log_{4} (2)}{\log_{16} (4)}$

Step 3

Rewrite $\log_{16} (4)$ as $\log_{16} (2^{2})$ .

$\frac{6 \log_{4} (2)}{\log_{16} (2^{2})}$

Step 4

Expand $\log_{16} (2^{2})$ by moving $2$ outside the logarithm.

$\frac{6 \log_{4} (2)}{2 \log_{16} (2)}$

Step 5

Cancel the common factor of $6$ and $2$ .

Tap for more steps...

Step 5.1

Factor $2$ out of $6 \log_{4} (2)$ .

$\frac{2 (3 \log_{4} (2))}{2 \log_{16} (2)}$

Step 5.2

Cancel the common factors.

Tap for more steps...

Step 5.2.1

Factor $2$ out of $2 \log_{16} (2)$ .

$\frac{2 (3 \log_{4} (2))}{2 (\log_{16} (2))}$

Step 5.2.2

Cancel the common factor.

$\frac{2 (3 \log_{4} (2))}{2 \log_{16} (2)}$

Step 5.2.3

Rewrite the expression.

$\frac{3 \log_{4} (2)}{\log_{16} (2)}$

Step 6

Simplify $3 \log_{4} (2)$ by moving $3$ inside the logarithm.

$\frac{\log_{4} (2^{3})}{\log_{16} (2)}$

Step 7

Logarithm base $16$ of $2$ is $\frac{1}{4}$ .

$\frac{\log_{4} (2^{3})}{\frac{1}{4}}$

Step 8

Simplify the numerator.

Tap for more steps...

Step 8.1

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

$\frac{\log_{4} (8)}{\frac{1}{4}}$

Step 8.2

Logarithm base $4$ of $8$ is $\frac{3}{2}$ .

$\frac{\frac{3}{2}}{\frac{1}{4}}$

Step 9

Multiply the numerator by the reciprocal of the denominator.

$\frac{3}{2} \cdot 4$

Step 10

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.1

Factor $2$ out of $4$ .

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

$3 \cdot 2$

Step 11

Multiply $3$ by $2$ .

$6$