Evaluate 1/3 log of 1000-2 log base 2 of 64- log base 6 of 6

$\frac{1}{3} \log (1000) - 2 \log_{2} (64) - \log_{6} (6)$

Simplify each term.

Logarithm base $10$ of $1000$ is $3$ .

$\frac{1}{3} \cdot 3 - 2 \log_{2} (64) - \log_{6} (6)$

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{1}{3} \cdot 3 - 2 \log_{2} (64) - \log_{6} (6)$

Rewrite the expression.

$1 - 2 \log_{2} (64) - \log_{6} (6)$

Logarithm base $2$ of $64$ is $6$ .

$1 - 2 \cdot 6 - \log_{6} (6)$

Multiply $- 2$ by $6$ .

$1 - 12 - \log_{6} (6)$

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

$1 - 12 - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$1 - 12 - 1$

Simplify by subtracting numbers.

Subtract $12$ from $1$ .

$- 11 - 1$

Subtract $1$ from $- 11$ .

$- 12$