Precalculus Examples

$\frac{\log_{3} (a^{3}) - 3 \log_{3} (b)}{\log_{9} (a b^{- 1})}$

Step 1

Simplify the numerator.

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Step 1.1

Simplify $- 3 \log_{3} (b)$ by moving $3$ inside the logarithm.

$\frac{\log_{3} (a^{3}) - \log_{3} (b^{3})}{\log_{9} (a b^{- 1})}$

Step 1.2

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

$\frac{\log_{3} (\frac{a^{3}}{b^{3}})}{\log_{9} (a b^{- 1})}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{\log_{3} (\frac{a^{3}}{b^{3}})}{\log_{9} (a \frac{1}{b})}$

Step 2.2

Combine $a$ and $\frac{1}{b}$ .

$\frac{\log_{3} (\frac{a^{3}}{b^{3}})}{\log_{9} (\frac{a}{b})}$

Step 3

Rewrite $\log_{3} (\frac{a^{3}}{b^{3}})$ using the change of base formula.

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Step 3.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}{\log_{9} (\frac{a}{b})}$

Step 3.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log (\frac{a^{3}}{b^{3}})}{\log (3)}}{\log_{9} (\frac{a}{b})}$

Step 4

Rewrite $\log_{9} (\frac{a}{b})$ using the change of base formula.

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Step 4.1

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\frac{\frac{\log (\frac{a^{3}}{b^{3}})}{\log (3)}}{\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)}}$

Step 4.2

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\frac{\log (\frac{a^{3}}{b^{3}})}{\log (3)}}{\frac{\log (\frac{a}{b})}{\log (9)}}$

Step 5

Multiply the numerator by the reciprocal of the denominator.

$\frac{\log (\frac{a^{3}}{b^{3}})}{\log (3)} \cdot \frac{\log (9)}{\log (\frac{a}{b})}$

Step 6

Combine.

$\frac{\log (\frac{a^{3}}{b^{3}}) \log (9)}{\log (3) \log (\frac{a}{b})}$