Precalculus Examples

$\ln (\frac{\sqrt[3]{x - 4}}{y^{2}})$

Step 1

Rewrite $\ln (\frac{\sqrt[3]{x - 4}}{y^{2}})$ as $\ln (\sqrt[3]{x - 4}) - \ln (y^{2})$ .

$\ln (\sqrt[3]{x - 4}) - \ln (y^{2})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x - 4}$ as ${(x - 4)}^{\frac{1}{3}}$ .

$\ln ({(x - 4)}^{\frac{1}{3}}) - \ln (y^{2})$

Step 3

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

$\frac{1}{3} \ln (x - 4) - \ln (y^{2})$

Step 4

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

$\frac{1}{3} \ln (x - 4) - (2 \ln (y))$

Step 5

Multiply $2$ by $- 1$ .

$\frac{1}{3} \ln (x - 4) - 2 \ln (y)$

Step 6

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

$\frac{\ln (x - 4)}{3} - 2 \ln (y)$

Step 7

To write $- 2 \ln (y)$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\ln (x - 4)}{3} - 2 \ln (y) \cdot \frac{3}{3}$

Step 8

Combine $- 2 \ln (y)$ and $\frac{3}{3}$ .

$\frac{\ln (x - 4)}{3} + \frac{- 2 \ln (y) \cdot 3}{3}$

Step 9

Combine the numerators over the common denominator.

$\frac{\ln (x - 4) - 2 \ln (y) \cdot 3}{3}$

Step 10

Multiply $3$ by $- 2$ .

$\frac{\ln (x - 4) - 6 \ln (y)}{3}$