Precalculus Examples

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

Step 1

Rewrite $\frac{1 \sqrt{3}}{3}$ as $1 \sqrt{3} \cdot 3^{- 1}$ .

$\log_{3} (1 \sqrt{3} \cdot 3^{- 1})$

Step 2

Rewrite $\log_{3} (1 \sqrt{3} \cdot 3^{- 1})$ as $\log_{3} (1 \sqrt{3}) + \log_{3} (3^{- 1})$ .

$\log_{3} (1 \sqrt{3}) + \log_{3} (3^{- 1})$

Step 3

Use logarithm rules to move $- 1$ out of the exponent.

$\log_{3} (1 \sqrt{3}) - \log_{3} (3)$

Step 4

Logarithm base $3$ of $3$ is $1$ .

$\log_{3} (1 \sqrt{3}) - 1 \cdot 1$

Step 5

Multiply $- 1$ by $1$ .

$\log_{3} (1 \sqrt{3}) - 1$

Step 6

Simplify each term.

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Step 6.1

Rewrite $\log_{3} (1 \sqrt{3})$ as $\log_{3} (1) + \log_{3} (\sqrt{3})$ .

$\log_{3} (1) + \log_{3} (\sqrt{3}) - 1$

Step 6.2

Logarithm base $3$ of $1$ is $0$ .

$0 + \log_{3} (\sqrt{3}) - 1$

Step 6.3

Add $0$ and $\log_{3} (\sqrt{3})$ .

$\log_{3} (\sqrt{3}) - 1$

Step 7

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

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Step 7.1

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 $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - 1$

Step 7.2

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

$\frac{\log (\sqrt{3})}{\log (3)} - 1$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt{3})}{\log (3)} - 1$

Decimal Form:

$- 0.5$