Precalculus Examples

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

Step 1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\log_{3} (\frac{3}{2} - 1 \cdot \frac{2}{2})$

Step 2

Combine $- 1$ and $\frac{2}{2}$ .

$\log_{3} (\frac{3}{2} + \frac{- 1 \cdot 2}{2})$

Step 3

Combine the numerators over the common denominator.

$\log_{3} (\frac{3 - 1 \cdot 2}{2})$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $- 1$ by $2$ .

$\log_{3} (\frac{3 - 2}{2})$

Step 4.2

Subtract $2$ from $3$ .

$\log_{3} (\frac{1}{2})$

Step 5

Rewrite $\log_{3} (\frac{1}{2})$ using the change of base formula.

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Step 5.1

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 5.2

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

$\frac{\log (\frac{1}{2})}{\log (3)}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\frac{1}{2})}{\log (3)}$

Decimal Form:

$- 0.63092975 \dots$