Precalculus Examples

$\frac{1}{3} \cdot (\log (a^{2}) + \log (b) - \log (c))$

Step 1

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

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

Step 2

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

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

Step 3

Simplify $\frac{1}{3} \log (\frac{a^{2} b}{c})$ by moving $\frac{1}{3}$ inside the logarithm.

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

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 $\frac{a^{2} b}{c}$ .

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

Step 4.2

Apply the product rule to $a^{2} b$ .

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

Step 5

Multiply the exponents in ${(a^{2})}^{\frac{1}{3}}$ .

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Step 5.1

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

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

Step 5.2

Combine $2$ and $\frac{1}{3}$ .

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