Precalculus Examples

$\log_{8} ({(\frac{y^{4} \sqrt{x}}{3 z^{2}})}^{3})$

Step 1

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

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Step 1.1

Apply the product rule to $\frac{y^{4} \sqrt{x}}{3 z^{2}}$ .

$\log_{8} (\frac{{(y^{4} \sqrt{x})}^{3}}{{(3 z^{2})}^{3}})$

Step 1.2

Apply the product rule to $y^{4} \sqrt{x}$ .

$\log_{8} (\frac{{(y^{4})}^{3} {\sqrt{x}}^{3}}{{(3 z^{2})}^{3}})$

Step 1.3

Apply the product rule to $3 z^{2}$ .

$\log_{8} (\frac{{(y^{4})}^{3} {\sqrt{x}}^{3}}{3^{3} {(z^{2})}^{3}})$

Step 2

Simplify the numerator.

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Step 2.1

Multiply the exponents in ${(y^{4})}^{3}$ .

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Step 2.1.1

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

$\log_{8} (\frac{y^{4 \cdot 3} {\sqrt{x}}^{3}}{3^{3} {(z^{2})}^{3}})$

Step 2.1.2

Multiply $4$ by $3$ .

$\log_{8} (\frac{y^{12} {\sqrt{x}}^{3}}{3^{3} {(z^{2})}^{3}})$

Step 2.2

Rewrite ${\sqrt{x}}^{3}$ as $\sqrt{x^{3}}$ .

$\log_{8} (\frac{y^{12} \sqrt{x^{3}}}{3^{3} {(z^{2})}^{3}})$

Step 2.3

Factor out $x^{2}$ .

$\log_{8} (\frac{y^{12} \sqrt{x^{2} x}}{3^{3} {(z^{2})}^{3}})$

Step 2.4

Pull terms out from under the radical.

$\log_{8} (\frac{y^{12} x \sqrt{x}}{3^{3} {(z^{2})}^{3}})$

Step 3

Simplify the denominator.

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Step 3.1

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

$\log_{8} (\frac{y^{12} x \sqrt{x}}{27 {(z^{2})}^{3}})$

Step 3.2

Multiply the exponents in ${(z^{2})}^{3}$ .

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Step 3.2.1

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

$\log_{8} (\frac{y^{12} x \sqrt{x}}{27 z^{2 \cdot 3}})$

Step 3.2.2

Multiply $2$ by $3$ .

$\log_{8} (\frac{y^{12} x \sqrt{x}}{27 z^{6}})$