Precalculus Examples

$\log (\frac{1}{x^{8}}) - 12 \log (x)$

Step 1

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

$\log (\frac{1}{x^{8}}) - \log (x^{12})$

Step 2

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

$\log (\frac{\frac{1}{x^{8}}}{x^{12}})$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\log (\frac{1}{x^{8}} \cdot \frac{1}{x^{12}})$

Step 4

Combine.

$\log (\frac{1 \cdot 1}{x^{8} x^{12}})$

Step 5

Multiply $x^{8}$ by $x^{12}$ by adding the exponents.

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Step 5.1

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

$\log (\frac{1 \cdot 1}{x^{8 + 12}})$

Step 5.2

Add $8$ and $12$ .

$\log (\frac{1 \cdot 1}{x^{20}})$

Step 6

Multiply $1$ by $1$ .

$\log (\frac{1}{x^{20}})$