Precalculus Examples

$\log_{3} (\frac{3}{12}) - \log_{3} (\frac{3}{4}) = x$

Step 1

Rewrite the equation as $x = \log_{3} (\frac{3}{12}) - \log_{3} (\frac{3}{4})$ .

$x = \log_{3} (\frac{3}{12}) - \log_{3} (\frac{3}{4})$

Step 2

Simplify $\log_{3} (\frac{3}{12}) - \log_{3} (\frac{3}{4})$ .

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Step 2.1

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

$x = \log_{3} (\frac{\frac{3}{12}}{\frac{3}{4}})$

Step 2.2

Cancel the common factor of $3$ and $12$ .

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Step 2.2.1

Factor $3$ out of $3$ .

$x = \log_{3} (\frac{\frac{3 (1)}{12}}{\frac{3}{4}})$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Factor $3$ out of $12$ .

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 2.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.4

Cancel the common factor of $4$ .

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Step 2.4.1

Cancel the common factor.

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

Step 2.4.2

Rewrite the expression.

$x = \log_{3} (\frac{1}{3})$

Step 2.5

Logarithm base $3$ of $\frac{1}{3}$ is $- 1$ .

$x = - 1$