Precalculus Examples

$\log_{4} (\frac{a^{3} b^{2}}{\sqrt[3]{c^{2}}})$

Step 1

Multiply $\frac{a^{3} b^{2}}{\sqrt[3]{c^{2}}}$ by $\frac{{\sqrt[3]{c^{2}}}^{2}}{{\sqrt[3]{c^{2}}}^{2}}$ .

$\log_{4} (\frac{a^{3} b^{2}}{\sqrt[3]{c^{2}}} \cdot \frac{{\sqrt[3]{c^{2}}}^{2}}{{\sqrt[3]{c^{2}}}^{2}})$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{a^{3} b^{2}}{\sqrt[3]{c^{2}}}$ by $\frac{{\sqrt[3]{c^{2}}}^{2}}{{\sqrt[3]{c^{2}}}^{2}}$ .

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{\sqrt[3]{c^{2}} {\sqrt[3]{c^{2}}}^{2}})$

Step 2.2

Raise $\sqrt[3]{c^{2}}$ to the power of $1$ .

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{{\sqrt[3]{c^{2}}}^{1} {\sqrt[3]{c^{2}}}^{2}})$

Step 2.3

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

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{{\sqrt[3]{c^{2}}}^{1 + 2}})$

Step 2.4

Add $1$ and $2$ .

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{{\sqrt[3]{c^{2}}}^{3}})$

Step 2.5

Rewrite ${\sqrt[3]{c^{2}}}^{3}$ as $c^{2}$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{c^{2}}$ as $c^{\frac{2}{3}}$ .

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{{(c^{\frac{2}{3}})}^{3}})$

Step 2.5.2

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

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{2}{3} \cdot 3}})$

Step 2.5.3

Combine $\frac{2}{3}$ and $3$ .

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{2 \cdot 3}{3}}})$

Step 2.5.4

Multiply $2$ by $3$ .

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{6}{3}}})$

Step 2.5.5

Cancel the common factor of $6$ and $3$ .

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Step 2.5.5.1

Factor $3$ out of $6$ .

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{3 \cdot 2}{3}}})$

Step 2.5.5.2

Cancel the common factors.

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Step 2.5.5.2.1

Factor $3$ out of $3$ .

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{3 \cdot 2}{3 (1)}}})$

Step 2.5.5.2.2

Cancel the common factor.

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{3 \cdot 2}{3 \cdot 1}}})$

Step 2.5.5.2.3

Rewrite the expression.

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{c^{\frac{2}{1}}})$

Step 2.5.5.2.4

Divide $2$ by $1$ .

$\log_{4} (\frac{a^{3} b^{2} {\sqrt[3]{c^{2}}}^{2}}{c^{2}})$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite ${\sqrt[3]{c^{2}}}^{2}$ as $\sqrt[3]{{(c^{2})}^{2}}$ .

$\log_{4} (\frac{a^{3} b^{2} \sqrt[3]{{(c^{2})}^{2}}}{c^{2}})$

Step 3.2

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

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Step 3.2.1

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

$\log_{4} (\frac{a^{3} b^{2} \sqrt[3]{c^{2 \cdot 2}}}{c^{2}})$

Step 3.2.2

Multiply $2$ by $2$ .

$\log_{4} (\frac{a^{3} b^{2} \sqrt[3]{c^{4}}}{c^{2}})$

Step 3.3

Factor out $c^{3}$ .

$\log_{4} (\frac{a^{3} b^{2} \sqrt[3]{c^{3} c}}{c^{2}})$

Step 3.4

Pull terms out from under the radical.

$\log_{4} (\frac{a^{3} b^{2} c \sqrt[3]{c}}{c^{2}})$

Step 4

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

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Step 4.1

Factor $c$ out of $a^{3} b^{2} c \sqrt[3]{c}$ .

$\log_{4} (\frac{c (a^{3} b^{2} \sqrt[3]{c})}{c^{2}})$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $c$ out of $c^{2}$ .

$\log_{4} (\frac{c (a^{3} b^{2} \sqrt[3]{c})}{c \cdot c})$

Step 4.2.2

Cancel the common factor.

$\log_{4} (\frac{c (a^{3} b^{2} \sqrt[3]{c})}{c \cdot c})$

Step 4.2.3

Rewrite the expression.

$\log_{4} (\frac{a^{3} b^{2} \sqrt[3]{c}}{c})$

Step 5

Rewrite $\log_{4} (\frac{a^{3} b^{2} \sqrt[3]{c}}{c})$ using the change of base formula.

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Step 5.1

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 5.2

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

$\frac{\log (\frac{a^{3} b^{2} \sqrt[3]{c}}{c})}{\log (4)}$