Precalculus Examples

$\log_{b} (\sqrt[4]{\frac{x^{5} (y^{2})}{w^{2}}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{\frac{x^{5} (y^{2})}{w^{2}}}$ as ${(\frac{x^{5} (y^{2})}{w^{2}})}^{\frac{1}{4}}$ .

$\log_{b} ({(\frac{x^{5} (y^{2})}{w^{2}})}^{\frac{1}{4}})$

Step 2

Expand $\log_{b} ({(\frac{x^{5} (y^{2})}{w^{2}})}^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$\frac{1}{4} \log_{b} (\frac{x^{5} (y^{2})}{w^{2}})$

Step 3

Multiply $x^{5}$ by $y^{2}$ .

$\frac{1}{4} \log_{b} (\frac{x^{5} y^{2}}{w^{2}})$

Step 4

Rewrite $\log_{b} (\frac{x^{5} y^{2}}{w^{2}})$ as $\log_{b} (x^{5} y^{2}) - \log_{b} (w^{2})$ .

$\frac{1}{4} (\log_{b} (x^{5} y^{2}) - \log_{b} (w^{2}))$

Step 5

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

$\frac{1}{4} (\log_{b} (x^{5} y^{2}) - (2 \log_{b} (w)))$

Step 6

Multiply $2$ by $- 1$ .

$\frac{1}{4} (\log_{b} (x^{5} y^{2}) - 2 \log_{b} (w))$

Step 7

Rewrite $\log_{b} (x^{5} y^{2})$ as $\log_{b} (x^{5}) + \log_{b} (y^{2})$ .

$\frac{1}{4} (\log_{b} (x^{5}) + \log_{b} (y^{2}) - 2 \log_{b} (w))$

Step 8

Expand $\log_{b} (x^{5})$ by moving $5$ outside the logarithm.

$\frac{1}{4} (5 \log_{b} (x) + \log_{b} (y^{2}) - 2 \log_{b} (w))$

Step 9

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

$\frac{1}{4} (5 \log_{b} (x) + 2 \log_{b} (y) - 2 \log_{b} (w))$

Step 10

Apply the distributive property.

$\frac{1}{4} (5 \log_{b} (x)) + \frac{1}{4} (2 \log_{b} (y)) + \frac{1}{4} (- 2 \log_{b} (w))$

Step 11

Simplify.

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Step 11.1

Multiply $\frac{1}{4} (5 \log_{b} (x))$ .

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Step 11.1.1

Combine $5$ and $\frac{1}{4}$ .

$\frac{5}{4} \log_{b} (x) + \frac{1}{4} (2 \log_{b} (y)) + \frac{1}{4} (- 2 \log_{b} (w))$

Step 11.1.2

Combine $\frac{5}{4}$ and $\log_{b} (x)$ .

$\frac{5 \log_{b} (x)}{4} + \frac{1}{4} (2 \log_{b} (y)) + \frac{1}{4} (- 2 \log_{b} (w))$

Step 11.2

Cancel the common factor of $2$ .

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Step 11.2.1

Factor $2$ out of $4$ .

$\frac{5 \log_{b} (x)}{4} + \frac{1}{2 (2)} (2 \log_{b} (y)) + \frac{1}{4} (- 2 \log_{b} (w))$

Step 11.2.2

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

$\frac{5 \log_{b} (x)}{4} + \frac{1}{2 (2)} (2 (\log_{b} (y))) + \frac{1}{4} (- 2 \log_{b} (w))$

Step 11.2.3

Cancel the common factor.

$\frac{5 \log_{b} (x)}{4} + \frac{1}{2 \cdot 2} (2 \log_{b} (y)) + \frac{1}{4} (- 2 \log_{b} (w))$

Step 11.2.4

Rewrite the expression.

$\frac{5 \log_{b} (x)}{4} + \frac{1}{2} \log_{b} (y) + \frac{1}{4} (- 2 \log_{b} (w))$

Step 11.3

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

$\frac{5 \log_{b} (x)}{4} + \frac{\log_{b} (y)}{2} + \frac{1}{4} (- 2 \log_{b} (w))$

Step 11.4

Cancel the common factor of $2$ .

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Step 11.4.1

Factor $2$ out of $4$ .

$\frac{5 \log_{b} (x)}{4} + \frac{\log_{b} (y)}{2} + \frac{1}{2 (2)} (- 2 \log_{b} (w))$

Step 11.4.2

Factor $2$ out of $- 2 \log_{b} (w)$ .

$\frac{5 \log_{b} (x)}{4} + \frac{\log_{b} (y)}{2} + \frac{1}{2 (2)} (2 (- \log_{b} (w)))$

Step 11.4.3

Cancel the common factor.

$\frac{5 \log_{b} (x)}{4} + \frac{\log_{b} (y)}{2} + \frac{1}{2 \cdot 2} (2 (- \log_{b} (w)))$

Step 11.4.4

Rewrite the expression.

$\frac{5 \log_{b} (x)}{4} + \frac{\log_{b} (y)}{2} + \frac{1}{2} (- \log_{b} (w))$

Step 11.5

Combine $\frac{1}{2}$ and $\log_{b} (w)$ .

$\frac{5 \log_{b} (x)}{4} + \frac{\log_{b} (y)}{2} - \frac{\log_{b} (w)}{2}$