Precalculus Examples

$\log_{3} (x^{- 6}) - \log_{3} (x^{- 2})$

Step 1

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

$\log_{3} (\frac{x^{- 6}}{x^{- 2}})$

Step 2

Move $x^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\log_{3} (\frac{1}{x^{- 2} x^{6}})$

Step 3

Multiply $x^{- 2}$ by $x^{6}$ by adding the exponents.

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Step 3.1

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

$\log_{3} (\frac{1}{x^{- 2 + 6}})$

Step 3.2

Add $- 2$ and $6$ .

$\log_{3} (\frac{1}{x^{4}})$

Step 4

Rewrite $\log_{3} (\frac{1}{x^{4}})$ 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 $0$ .

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

Step 4.2

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

$\frac{\log (\frac{1}{x^{4}})}{\log (3)}$