Precalculus Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

Raise $\sqrt[4]{x}$ to the power of $1$ .

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

Step 2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.4

Add $1$ and $3$ .

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

Step 2.5

Rewrite ${\sqrt[4]{x}}^{4}$ as $x$ .

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Step 2.5.1

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

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

Step 2.5.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.5.3

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

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

Step 2.5.4

Cancel the common factor of $4$ .

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Step 2.5.4.1

Cancel the common factor.

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

Step 2.5.4.2

Rewrite the expression.

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

Step 2.5.5

Simplify.

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

Step 3

Rewrite ${\sqrt[4]{x}}^{3}$ as $\sqrt[4]{x^{3}}$ .

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

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

Apply the product rule to $\frac{8 y (x - 5) \sqrt[4]{x^{3}}}{x}$ .

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

Step 4.2

Apply the product rule to $8 y (x - 5) \sqrt[4]{x^{3}}$ .

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

Step 4.3

Apply the product rule to $8 y (x - 5)$ .

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

Step 4.4

Apply the product rule to $8 y$ .

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

Step 5

Simplify the numerator.

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Step 5.1

Raise $8$ to the power of $3$ .

$\log_{8} (\frac{512 y^{3} {(x - 5)}^{3} {\sqrt[4]{x^{3}}}^{3}}{x^{3}})$

Step 5.2

Rewrite ${\sqrt[4]{x^{3}}}^{3}$ as $\sqrt[4]{{(x^{3})}^{3}}$ .

$\log_{8} (\frac{512 y^{3} {(x - 5)}^{3} \sqrt[4]{{(x^{3})}^{3}}}{x^{3}})$

Step 5.3

Multiply the exponents in ${(x^{3})}^{3}$ .

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Step 5.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{8} (\frac{512 y^{3} {(x - 5)}^{3} \sqrt[4]{x^{3 \cdot 3}}}{x^{3}})$

Step 5.3.2

Multiply $3$ by $3$ .

$\log_{8} (\frac{512 y^{3} {(x - 5)}^{3} \sqrt[4]{x^{9}}}{x^{3}})$

Step 5.4

Rewrite $x^{9}$ as ${(x^{2})}^{4} x$ .

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Step 5.4.1

Factor out $x^{8}$ .

$\log_{8} (\frac{512 y^{3} {(x - 5)}^{3} \sqrt[4]{x^{8} x}}{x^{3}})$

Step 5.4.2

Rewrite $x^{8}$ as ${(x^{2})}^{4}$ .

$\log_{8} (\frac{512 y^{3} {(x - 5)}^{3} \sqrt[4]{{(x^{2})}^{4} x}}{x^{3}})$

Step 5.5

Pull terms out from under the radical.

$\log_{8} (\frac{512 y^{3} {(x - 5)}^{3} x^{2} \sqrt[4]{x}}{x^{3}})$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $x^{2}$ and $x^{3}$ .

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Step 6.1.1

Factor $x^{2}$ out of $512 y^{3} {(x - 5)}^{3} x^{2} \sqrt[4]{x}$ .

$\log_{8} (\frac{x^{2} (512 y^{3} {(x - 5)}^{3} \sqrt[4]{x})}{x^{3}})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $x^{2}$ out of $x^{3}$ .

$\log_{8} (\frac{x^{2} (512 y^{3} {(x - 5)}^{3} \sqrt[4]{x})}{x^{2} x})$

Step 6.1.2.2

Cancel the common factor.

$\log_{8} (\frac{x^{2} (512 y^{3} {(x - 5)}^{3} \sqrt[4]{x})}{x^{2} x})$

Step 6.1.2.3

Rewrite the expression.

$\log_{8} (\frac{512 y^{3} {(x - 5)}^{3} \sqrt[4]{x}}{x})$

Step 6.2

Reorder factors in $\log_{8} (\frac{512 y^{3} {(x - 5)}^{3} \sqrt[4]{x}}{x})$ .

$\log_{8} (\frac{512 y^{3} \sqrt[4]{x} {(x - 5)}^{3}}{x})$

Step 7

Rewrite $\log_{8} (\frac{512 y^{3} \sqrt[4]{x} {(x - 5)}^{3}}{x})$ using the change of base formula.

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Step 7.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}$

Step 7.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (\frac{512 y^{3} \sqrt[4]{x} {(x - 5)}^{3}}{x})}{\log (8)}$