Precalculus Examples

$\ln (\frac{\sqrt[3]{x}}{10 \sqrt{y z}})$

Step 1

Rewrite $\ln (\frac{\sqrt[3]{x}}{10 \sqrt{y z}})$ as $\ln (\sqrt[3]{x}) - \ln (10 \sqrt{y z})$ .

$\ln (\sqrt[3]{x}) - \ln (10 \sqrt{y z})$

Step 2

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

$\ln (x^{\frac{1}{3}}) - \ln (10 \sqrt{y z})$

Step 3

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

$\ln (x^{\frac{1}{3}}) - \ln (10 {(y z)}^{\frac{1}{2}})$

Step 4

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

$\frac{1}{3} \ln (x) - \ln (10 {(y z)}^{\frac{1}{2}})$

Step 5

Rewrite $\ln (10 {(y z)}^{\frac{1}{2}})$ as $\ln (10) + \ln ({(y z)}^{\frac{1}{2}})$ .

$\frac{1}{3} \ln (x) - (\ln (10) + \ln ({(y z)}^{\frac{1}{2}}))$

Step 6

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

$\frac{1}{3} \ln (x) - (\ln (10) + \frac{1}{2} \ln (y z))$

Step 7

Rewrite $\ln (y z)$ as $\ln (y) + \ln (z)$ .

$\frac{1}{3} \ln (x) - (\ln (10) + \frac{1}{2} (\ln (y) + \ln (z)))$

Step 8

Simplify each term.

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Step 8.1

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

$\frac{\ln (x)}{3} - (\ln (10) + \frac{1}{2} (\ln (y) + \ln (z)))$

Step 8.2

Simplify each term.

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Step 8.2.1

Rewrite $\ln (10)$ as $\ln (2 (5))$ .

$\frac{\ln (x)}{3} - (\ln (2 (5)) + \frac{1}{2} (\ln (y) + \ln (z)))$

Step 8.2.2

Rewrite $\ln (2 (5))$ as $\ln (2) + \ln (5)$ .

$\frac{\ln (x)}{3} - (\ln (2) + \ln (5) + \frac{1}{2} (\ln (y) + \ln (z)))$

Step 8.2.3

Apply the distributive property.

$\frac{\ln (x)}{3} - (\ln (2) + \ln (5) + \frac{1}{2} \ln (y) + \frac{1}{2} \ln (z))$

Step 8.2.4

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

$\frac{\ln (x)}{3} - (\ln (2) + \ln (5) + \frac{\ln (y)}{2} + \frac{1}{2} \ln (z))$

Step 8.2.5

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

$\frac{\ln (x)}{3} - (\ln (2) + \ln (5) + \frac{\ln (y)}{2} + \frac{\ln (z)}{2})$

Step 8.3

Apply the distributive property.

$\frac{\ln (x)}{3} - \ln (2) - \ln (5) - \frac{\ln (y)}{2} - \frac{\ln (z)}{2}$