Evaluate log base 6 of 3- log base 6 of 18

$\log_{6} (3) - \log_{6} (18)$

Step 1

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

$\log_{6} (\frac{3}{18})$

Step 2

Reduce the expression $\frac{3}{18}$ by cancelling the common factors.

Factor $3$ out of $3$ .

$\log_{6} (\frac{3 (1)}{18})$

Factor $3$ out of $18$ .

$\log_{6} (\frac{3 \cdot 1}{3 \cdot 6})$

Cancel the common factor.

$\log_{6} (\frac{3 \cdot 1}{3 \cdot 6})$

Rewrite the expression.

$\log_{6} (\frac{1}{6})$

Step 3

Rewrite $\frac{1}{6}$ as $1 \cdot 6^{- 1}$ .

$\log_{6} (1 \cdot 6^{- 1})$

Step 4

Rewrite $\log_{6} (1 \cdot 6^{- 1})$ as $\log_{6} (1) + \log_{6} (6^{- 1})$ .

$\log_{6} (1) + \log_{6} (6^{- 1})$

Step 5

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

$\log_{6} (1) - \log_{6} (6)$

Step 6

Logarithm base $6$ of $6$ is $1$ .

$\log_{6} (1) - 1 \cdot 1$

Step 7

Multiply $- 1$ by $1$ .

$\log_{6} (1) - 1$

Step 8

Logarithm base $6$ of $1$ is $0$ .

$0 - 1$

Step 9

Subtract $1$ from $0$ .

$- 1$