Precalculus Examples

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

Step 1

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

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

Step 2

Cancel the common factor of $3^{x - 6}$ and $3$ .

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Step 2.1

Factor $3$ out of $3^{x - 6}$ .

$\log_{3} (\frac{3 \cdot 3^{x - 7}}{3 x})$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $3$ out of $3 x$ .

$\log_{3} (\frac{3 \cdot 3^{x - 7}}{3 (x)})$

Step 2.2.2

Cancel the common factor.

$\log_{3} (\frac{3 \cdot 3^{x - 7}}{3 x})$

Step 2.2.3

Rewrite the expression.

$\log_{3} (\frac{3^{x - 7}}{x})$

Step 3

Rewrite $\log_{3} (\frac{3^{x - 7}}{x})$ 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 $0$ .

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

Step 3.2

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

$\frac{\log (\frac{3^{x - 7}}{x})}{\log (3)}$