Precalculus Examples

$\log_{15} (\sqrt[4]{\frac{11}{y^{2} x}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{\frac{11}{y^{2} x}}$ as ${(\frac{11}{y^{2} x})}^{\frac{1}{4}}$ .

$\log_{15} ({(\frac{11}{y^{2} x})}^{\frac{1}{4}})$

Step 2

Expand $\log_{15} ({(\frac{11}{y^{2} x})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{15} (\frac{11}{y^{2} x})$

Step 3

Rewrite $\log_{15} (\frac{11}{y^{2} x})$ as $\log_{15} (11) - \log_{15} (y^{2} x)$ .

$\frac{1}{4} (\log_{15} (11) - \log_{15} (y^{2} x))$

Step 4

Rewrite $\log_{15} (y^{2} x)$ as $\log_{15} (y^{2}) + \log_{15} (x)$ .

$\frac{1}{4} (\log_{15} (11) - (\log_{15} (y^{2}) + \log_{15} (x)))$

Step 5

Expand $\log_{15} (y^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{4} (\log_{15} (11) - (2 \log_{15} (y) + \log_{15} (x)))$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$\frac{1}{4} (\log_{15} (11) - (2 \log_{15} (y)) - \log_{15} (x))$

Step 6.2

Multiply $2$ by $- 1$ .

$\frac{1}{4} (\log_{15} (11) - 2 \log_{15} (y) - \log_{15} (x))$

Step 7

Apply the distributive property.

$\frac{1}{4} \log_{15} (11) + \frac{1}{4} (- 2 \log_{15} (y)) + \frac{1}{4} (- \log_{15} (x))$

Step 8

Simplify.

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Step 8.1

Combine $\frac{1}{4}$ and $\log_{15} (11)$ .

$\frac{\log_{15} (11)}{4} + \frac{1}{4} (- 2 \log_{15} (y)) + \frac{1}{4} (- \log_{15} (x))$

Step 8.2

Cancel the common factor of $2$ .

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Step 8.2.1

Factor $2$ out of $4$ .

$\frac{\log_{15} (11)}{4} + \frac{1}{2 (2)} (- 2 \log_{15} (y)) + \frac{1}{4} (- \log_{15} (x))$

Step 8.2.2

Factor $2$ out of $- 2 \log_{15} (y)$ .

$\frac{\log_{15} (11)}{4} + \frac{1}{2 (2)} (2 (- \log_{15} (y))) + \frac{1}{4} (- \log_{15} (x))$

Step 8.2.3

Cancel the common factor.

$\frac{\log_{15} (11)}{4} + \frac{1}{2 \cdot 2} (2 (- \log_{15} (y))) + \frac{1}{4} (- \log_{15} (x))$

Step 8.2.4

Rewrite the expression.

$\frac{\log_{15} (11)}{4} + \frac{1}{2} (- \log_{15} (y)) + \frac{1}{4} (- \log_{15} (x))$

Step 8.3

Combine $\frac{1}{2}$ and $\log_{15} (y)$ .

$\frac{\log_{15} (11)}{4} - \frac{\log_{15} (y)}{2} + \frac{1}{4} (- \log_{15} (x))$

Step 8.4

Combine $\frac{1}{4}$ and $\log_{15} (x)$ .

$\frac{\log_{15} (11)}{4} - \frac{\log_{15} (y)}{2} - \frac{\log_{15} (x)}{4}$