Precalculus Examples

$\frac{1}{5} \cdot \log (a^{5}) - 12 \log (b) + \log (c^{12}) - \log (d)$

Step 1

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

$\frac{1}{5} \cdot \log (a^{5}) - 12 \log (b) + \log (\frac{c^{12}}{d})$

Step 2

Simplify each term.

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Step 2.1

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

$\log ({(a^{5})}^{\frac{1}{5}}) - 12 \log (b) + \log (\frac{c^{12}}{d})$

Step 2.2

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

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Step 2.2.1

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

$\log (a^{5 (\frac{1}{5})}) - 12 \log (b) + \log (\frac{c^{12}}{d})$

Step 2.2.2

Cancel the common factor of $5$ .

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Step 2.2.2.1

Cancel the common factor.

$\log (a^{5 (\frac{1}{5})}) - 12 \log (b) + \log (\frac{c^{12}}{d})$

Step 2.2.2.2

Rewrite the expression.

$\log (a^{1}) - 12 \log (b) + \log (\frac{c^{12}}{d})$

Step 2.3

Simplify.

$\log (a) - 12 \log (b) + \log (\frac{c^{12}}{d})$

Step 2.4

Simplify $- 12 \log (b)$ by moving $12$ inside the logarithm.

$\log (a) - \log (b^{12}) + \log (\frac{c^{12}}{d})$

Step 3

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

$\log (\frac{a}{b^{12}}) + \log (\frac{c^{12}}{d})$

Step 4

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

$\log (\frac{a}{b^{12}} \cdot \frac{c^{12}}{d})$

Step 5

Combine.

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