Precalculus Examples

$\log (\sqrt[4]{\frac{x^{2}}{y^{2}}})$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Multiply $2$ by $- 1$ .

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

Step 7

Apply the distributive property.

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

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Factor $2$ out of $4$ .

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

Step 8.2

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

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

Step 8.3

Cancel the common factor.

$\frac{1}{2 \cdot 2} (2 \log (x)) + \frac{1}{4} (- 2 \log (y))$

Step 8.4

Rewrite the expression.

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

Step 9

Combine $\frac{1}{2}$ and $\log (x)$ .

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

Step 10

Cancel the common factor of $2$ .

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Step 10.1

Factor $2$ out of $4$ .

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

Step 10.2

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

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

Step 10.3

Cancel the common factor.

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

Step 10.4

Rewrite the expression.

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

Step 11

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

$\frac{\log (x)}{2} - \frac{\log (y)}{2}$