Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Combine $\frac{1}{2}$ and $\ln (y)$ .

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify each term.

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Step 9.1

Apply the distributive property.

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

Step 9.2

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

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

Step 9.3

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

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Step 9.3.1

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

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

Step 9.3.2

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

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