Precalculus Examples

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

Step 1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

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

Step 2

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

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

Step 3

Combine using the product rule for radicals.

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

Step 4

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

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

Step 5

Combine and simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $2$ .

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

Step 5.5

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

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Step 5.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{c}$ as $c^{\frac{1}{3}}$ .

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

Step 5.5.2

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

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

Step 5.5.3

Combine $\frac{1}{3}$ and $3$ .

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

Step 5.5.4

Cancel the common factor of $3$ .

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Step 5.5.4.1

Cancel the common factor.

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

Step 5.5.4.2

Rewrite the expression.

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

Step 5.5.5

Simplify.

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

Step 6

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

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

Step 7

Combine using the product rule for radicals.

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