Precalculus Examples

$\log_{4} (\frac{16 x^{2}}{\sqrt[3]{x - 1}})$

Step 1

Rewrite $\log_{4} (\frac{16 x^{2}}{\sqrt[3]{x - 1}})$ as $\log_{4} (16 x^{2}) - \log_{4} (\sqrt[3]{x - 1})$ .

$\log_{4} (16 x^{2}) - \log_{4} (\sqrt[3]{x - 1})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 1}$ as ${(x - 1)}^{\frac{1}{3}}$ .

$\log_{4} (16 x^{2}) - \log_{4} ({(x - 1)}^{\frac{1}{3}})$

Step 3

Expand $\log_{4} ({(x - 1)}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\log_{4} (16 x^{2}) - (\frac{1}{3} \log_{4} (x - 1))$

Step 4

Combine $\frac{1}{3}$ and $\log_{4} (x - 1)$ .

$\log_{4} (16 x^{2}) - \frac{\log_{4} (x - 1)}{3}$

Step 5

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

$\log_{4} (16) + \log_{4} (x^{2}) - \frac{\log_{4} (x - 1)}{3}$

Step 6

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

$\log_{4} (16) + 2 \log_{4} (x) - \frac{\log_{4} (x - 1)}{3}$

Step 7

Logarithm base $4$ of $16$ is $2$ .

$2 + 2 \log_{4} (x) - \frac{\log_{4} (x - 1)}{3}$

Step 8

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 \log_{4} (x) + 2 \cdot \frac{3}{3} - \frac{\log_{4} (x - 1)}{3}$

Step 9

Combine $2$ and $\frac{3}{3}$ .

$2 \log_{4} (x) + \frac{2 \cdot 3}{3} - \frac{\log_{4} (x - 1)}{3}$

Step 10

Combine the numerators over the common denominator.

$2 \log_{4} (x) + \frac{2 \cdot 3 - \log_{4} (x - 1)}{3}$

Step 11

Multiply $2$ by $3$ .

$2 \log_{4} (x) + \frac{6 - \log_{4} (x - 1)}{3}$