Precalculus Examples

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y^{3} z^{2}}$ as ${(y^{3} z^{2})}^{\frac{1}{3}}$ .

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

Step 3

Expand $\ln ({(y^{3} z^{2})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

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

Step 4

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

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

Step 5

Multiply $4$ by $- 1$ .

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

Step 6

Rewrite $\ln (y^{3} z^{2})$ as $\ln (y^{3}) + \ln (z^{2})$ .

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

Step 7

Expand $\ln (y^{3})$ by moving $3$ outside the logarithm.

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

Step 8

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

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

Step 9

Simplify each term.

Tap for more steps...

Step 9.1

Apply the distributive property.

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

Step 9.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 9.2.1

Factor $3$ out of $3 \ln (y)$ .

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

Step 9.2.2

Cancel the common factor.

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

Step 9.2.3

Rewrite the expression.

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

Step 9.3

Multiply $\frac{1}{3} (2 \ln (z))$ .

Tap for more steps...

Step 9.3.1

Combine $2$ and $\frac{1}{3}$ .

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

Step 9.3.2

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

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