Precalculus Examples

$\ln (\sqrt{\frac{x^{2}}{y^{2}}})$

Step 1

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

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

Step 2

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

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

Step 3

Rewrite $\ln (\frac{x^{2}}{y^{2}})$ as $\ln (x^{2}) - \ln (y^{2})$ .

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

Step 4

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

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

Step 5

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

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

Step 6

Multiply $2$ by $- 1$ .

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

Step 7

Apply the distributive property.

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

Step 8

Cancel the common factor of $2$ .

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Step 8.1

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

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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

Step 9

Cancel the common factor of $2$ .

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Step 9.1

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

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

Step 9.2

Cancel the common factor.

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

Step 9.3

Rewrite the expression.

$\ln (x) - \ln (y)$