Precalculus Examples

$\log_{8} ({(\frac{8 y (x - 5)}{\sqrt[4]{x}})}^{3})$

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

$\log_{8} ({(\frac{8 y (x - 5)}{x^{\frac{1}{4}}})}^{3})$

Expand $\log_{8} ({(\frac{8 y (x - 5)}{x^{\frac{1}{4}}})}^{3})$ by moving $3$ outside the logarithm.

$3 \log_{8} (\frac{8 y (x - 5)}{x^{\frac{1}{4}}})$

Rewrite $\log_{8} (\frac{8 y (x - 5)}{x^{\frac{1}{4}}})$ as $\log_{8} (8 y (x - 5)) - \log_{8} (x^{\frac{1}{4}})$ .

$3 (\log_{8} (8 y (x - 5)) - \log_{8} (x^{\frac{1}{4}}))$

Expand $\log_{8} (x^{\frac{1}{4}})$ by moving $\frac{1}{4}$ outside the logarithm.

$3 (\log_{8} (8 y (x - 5)) - (\frac{1}{4} \log_{8} (x)))$

Combine $\frac{1}{4}$ and $\log_{8} (x)$ .

$3 (\log_{8} (8 y (x - 5)) - \frac{\log_{8} (x)}{4})$

Rewrite $\log_{8} (8 y (x - 5))$ as $\log_{8} (8 y) + \log_{8} (x - 5)$ .

$3 (\log_{8} (8 y) + \log_{8} (x - 5) - \frac{\log_{8} (x)}{4})$

Rewrite $\log_{8} (8 y)$ as $\log_{8} (8) + \log_{8} (y)$ .

$3 (\log_{8} (8) + \log_{8} (y) + \log_{8} (x - 5) - \frac{\log_{8} (x)}{4})$

Logarithm base $8$ of $8$ is $1$ .

$3 (1 + \log_{8} (y) + \log_{8} (x - 5) - \frac{\log_{8} (x)}{4})$

To write $\log_{8} (x - 5)$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$3 (1 + \log_{8} (y) + \log_{8} (x - 5) \cdot \frac{4}{4} - \frac{\log_{8} (x)}{4})$

Combine $\log_{8} (x - 5)$ and $\frac{4}{4}$ .

$3 (1 + \log_{8} (y) + \frac{\log_{8} (x - 5) \cdot 4}{4} - \frac{\log_{8} (x)}{4})$

Combine the numerators over the common denominator.

$3 (1 + \log_{8} (y) + \frac{\log_{8} (x - 5) \cdot 4 - \log_{8} (x)}{4})$

Move $4$ to the left of $\log_{8} (x - 5)$ .

$3 (1 + \log_{8} (y) + \frac{4 \log_{8} (x - 5) - \log_{8} (x)}{4})$

Apply the distributive property.

$3 \cdot 1 + 3 \log_{8} (y) + 3 \frac{4 \log_{8} (x - 5) - \log_{8} (x)}{4}$

Simplify.

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Multiply $3$ by $1$ .

$3 + 3 \log_{8} (y) + 3 \frac{4 \log_{8} (x - 5) - \log_{8} (x)}{4}$

Combine $3$ and $\frac{4 \log_{8} (x - 5) - \log_{8} (x)}{4}$ .

$3 + 3 \log_{8} (y) + \frac{3 (4 \log_{8} (x - 5) - \log_{8} (x))}{4}$

To write $3$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$3 \log_{8} (y) + 3 \cdot \frac{4}{4} + \frac{3 (4 \log_{8} (x - 5) - \log_{8} (x))}{4}$

Combine $3$ and $\frac{4}{4}$ .

$3 \log_{8} (y) + \frac{3 \cdot 4}{4} + \frac{3 (4 \log_{8} (x - 5) - \log_{8} (x))}{4}$

Combine the numerators over the common denominator.

$3 \log_{8} (y) + \frac{3 \cdot 4 + 3 (4 \log_{8} (x - 5) - \log_{8} (x))}{4}$

Multiply $3$ by $4$ .

$3 \log_{8} (y) + \frac{12 + 3 (4 \log_{8} (x - 5) - \log_{8} (x))}{4}$

Factor $3$ out of $12 + 3 (4 \log_{8} (x - 5) - \log_{8} (x))$ .

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Factor $3$ out of $12$ .

$3 \log_{8} (y) + \frac{3 \cdot 4 + 3 (4 \log_{8} (x - 5) - \log_{8} (x))}{4}$

Factor $3$ out of $3 \cdot 4 + 3 (4 \log_{8} (x - 5) - \log_{8} (x))$ .

$3 \log_{8} (y) + \frac{3 (4 + 4 \log_{8} (x - 5) - \log_{8} (x))}{4}$

To write $3 \log_{8} (y)$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$3 \log_{8} (y) \cdot \frac{4}{4} + \frac{3 (4 + 4 \log_{8} (x - 5) - \log_{8} (x))}{4}$

Combine $3 \log_{8} (y)$ and $\frac{4}{4}$ .

$\frac{3 \log_{8} (y) \cdot 4}{4} + \frac{3 (4 + 4 \log_{8} (x - 5) - \log_{8} (x))}{4}$

Combine the numerators over the common denominator.

$\frac{3 \log_{8} (y) \cdot 4 + 3 (4 + 4 \log_{8} (x - 5) - \log_{8} (x))}{4}$

Simplify the numerator.

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Factor $3$ out of $3 \log_{8} (y) \cdot 4 + 3 (4 + 4 \log_{8} (x - 5) - \log_{8} (x))$ .

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Factor $3$ out of $3 \log_{8} (y) \cdot 4$ .

$\frac{3 (\log_{8} (y) \cdot 4) + 3 (4 + 4 \log_{8} (x - 5) - \log_{8} (x))}{4}$

Factor $3$ out of $3 (\log_{8} (y) \cdot 4) + 3 (4 + 4 \log_{8} (x - 5) - \log_{8} (x))$ .

$\frac{3 (\log_{8} (y) \cdot 4 + 4 + 4 \log_{8} (x - 5) - \log_{8} (x))}{4}$

Move $4$ to the left of $\log_{8} (y)$ .

$\frac{3 (4 \log_{8} (y) + 4 + 4 \log_{8} (x - 5) - \log_{8} (x))}{4}$