Precalculus Examples

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

Step 1

Apply the product rule to $\frac{1}{3}$ .

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

Step 2

Move $3^{2}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

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

Step 3

Rewrite $\log_{3} (1^{2} \cdot 3^{- 2})$ as $\log_{3} (1^{2}) + \log_{3} (3^{- 2})$ .

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

Step 4

Use logarithm rules to move $- 2$ out of the exponent.

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

Step 5

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

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

Step 6

Multiply $- 2$ by $1$ .

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

Step 7

Simplify each term.

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Step 7.1

Expand $\log_{3} (1^{2})$ by moving $2$ outside the logarithm.

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

Step 7.2

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

$2 \cdot 0 - 2$

Step 7.3

Multiply $2$ by $0$ .

$0 - 2$

Step 8

Subtract $2$ from $0$ .

$- 2$