Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Multiply $3$ by $- 1$ .

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

Step 7

Apply the distributive property.

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

Step 8

Cancel the common factor of $2$ .

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Step 8.1

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

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 10

Move the negative in front of the fraction.

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