Precalculus Examples

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

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{x^{3} y^{4}}{3 z}}$ as ${(\frac{x^{3} y^{4}}{3 z})}^{\frac{1}{2}}$ .

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

Step 2

Expand $\ln ({(\frac{x^{3} y^{4}}{3 z})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 3

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

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

Step 4

Rewrite $\ln (x^{3} y^{4})$ as $\ln (x^{3}) + \ln (y^{4})$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Rewrite $\ln (3 z)$ as $\ln (3) + \ln (z)$ .

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

Step 8

Apply the distributive property.

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

Step 9

Apply the distributive property.

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

Step 10

Simplify.

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Step 10.1

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

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Step 10.1.1

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

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

Step 10.1.2

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

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

Step 10.2

Cancel the common factor of $2$ .

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Step 10.2.1

Factor $2$ out of $4 \ln (y)$ .

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

Step 10.2.2

Cancel the common factor.

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

Step 10.2.3

Rewrite the expression.

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

Step 10.3

Combine $\frac{1}{2}$ and $\ln (3)$ .

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

Step 10.4

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

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