Precalculus Examples

$\frac{1}{3} \cdot (\frac{2 \log (a) + \log (b) - 5 \log (c)}{4})$

Step 1

Simplify the numerator.

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Step 1.1

Simplify $2 \log (a)$ by moving $2$ inside the logarithm.

$\frac{1}{3} \cdot \frac{\log (a^{2}) + \log (b) - 5 \log (c)}{4}$

Step 1.2

Simplify $- 5 \log (c)$ by moving $5$ inside the logarithm.

$\frac{1}{3} \cdot \frac{\log (a^{2}) + \log (b) - \log (c^{5})}{4}$

Step 1.3

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

$\frac{1}{3} \cdot \frac{\log (a^{2} b) - \log (c^{5})}{4}$

Step 1.4

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

$\frac{1}{3} \cdot \frac{\log (\frac{a^{2} b}{c^{5}})}{4}$

Step 2

Combine fractions.

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Step 2.1

Combine.

$\frac{1 \log (\frac{a^{2} b}{c^{5}})}{3 \cdot 4}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Multiply $\log (\frac{a^{2} b}{c^{5}})$ by $1$ .

$\frac{\log (\frac{a^{2} b}{c^{5}})}{3 \cdot 4}$

Step 2.2.2

Multiply $3$ by $4$ .

$\frac{\log (\frac{a^{2} b}{c^{5}})}{12}$

Step 2.2.3

Rewrite $\frac{\log (\frac{a^{2} b}{c^{5}})}{12}$ as $\frac{1}{12} \log (\frac{a^{2} b}{c^{5}})$ .

$\frac{1}{12} \log (\frac{a^{2} b}{c^{5}})$

Step 3

Simplify $\frac{1}{12} \log (\frac{a^{2} b}{c^{5}})$ by moving $\frac{1}{12}$ inside the logarithm.

$\log ({(\frac{a^{2} b}{c^{5}})}^{\frac{1}{12}})$

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 $\frac{a^{2} b}{c^{5}}$ .

$\log (\frac{{(a^{2} b)}^{\frac{1}{12}}}{{(c^{5})}^{\frac{1}{12}}})$

Step 4.2

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

$\log (\frac{{(a^{2})}^{\frac{1}{12}} b^{\frac{1}{12}}}{{(c^{5})}^{\frac{1}{12}}})$

Step 5

Multiply the exponents in ${(a^{2})}^{\frac{1}{12}}$ .

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Step 5.1

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

$\log (\frac{a^{2 (\frac{1}{12})} b^{\frac{1}{12}}}{{(c^{5})}^{\frac{1}{12}}})$

Step 5.2

Cancel the common factor of $2$ .

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Step 5.2.1

Factor $2$ out of $12$ .

$\log (\frac{a^{2 \frac{1}{2 (6)}} b^{\frac{1}{12}}}{{(c^{5})}^{\frac{1}{12}}})$

Step 5.2.2

Cancel the common factor.

$\log (\frac{a^{2 \frac{1}{2 \cdot 6}} b^{\frac{1}{12}}}{{(c^{5})}^{\frac{1}{12}}})$

Step 5.2.3

Rewrite the expression.

$\log (\frac{a^{\frac{1}{6}} b^{\frac{1}{12}}}{{(c^{5})}^{\frac{1}{12}}})$

Step 6

Multiply the exponents in ${(c^{5})}^{\frac{1}{12}}$ .

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Step 6.1

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

$\log (\frac{a^{\frac{1}{6}} b^{\frac{1}{12}}}{c^{5 (\frac{1}{12})}})$

Step 6.2

Combine $5$ and $\frac{1}{12}$ .

$\log (\frac{a^{\frac{1}{6}} b^{\frac{1}{12}}}{c^{\frac{5}{12}}})$