Algebra Examples

$\log_{3} (13) - \log_{3} (39)$

Step 1

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

$\log_{3} (\frac{13}{39})$

Step 2

Reduce the expression $\frac{13}{39}$ by cancelling the common factors.

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Step 2.1

Factor $13$ out of $13$ .

$\log_{3} (\frac{13 (1)}{39})$

Step 2.2

Factor $13$ out of $39$ .

$\log_{3} (\frac{13 \cdot 1}{13 \cdot 3})$

Step 2.3

Cancel the common factor.

$\log_{3} (\frac{13 \cdot 1}{13 \cdot 3})$

Step 2.4

Rewrite the expression.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Multiply $- 1$ by $1$ .

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

Step 8

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

$0 - 1$

Step 9

Subtract $1$ from $0$ .

$- 1$