Answer
$\frac{2}{3}$
Work Step by Step
$Evaluate$ $the$ $expression$ $without$ $using$ $a$ $calculator:$
$\log_8 6 - \log_8 3 + \log_8 2$
Use the Second Law of Logarithms for $\log_8 6 - \log_8 3$
$$\log_8 6 - \log_8 3 = \log_8 (\frac{6}{3})$$
$$\log_8 2 + \log_8 2$$
Use the First Law of Logarithms
$$\log_8 2 + \log_8 2 = \log_8 (2\times2)$$
$$\log_8 4$$
Use the First Law of Logarithms
$$\log_8 (8\times \frac{1}{2}) = \log_8 8 + \log_8 \frac{1}{2}$$
Rewrite $\frac{1}{2}$ as $8^{-\frac{1}{3}}$ [Note: $8^{-\frac{1}{3}} = \frac{1}{8^{\frac{1}{3}}} = \frac{1}{\sqrt[3] 8} = \frac{1}{2}$]
$$\log_8 8 + \log_8 8^{-\frac{1}{3}}$$
Use the Third Property of Logarithms: $\log_a a^x = x$
$$\log_8 8 + \log_8 8^{-\frac{1}{3}} = 1-\frac{1}{3}$$
$$\frac{3}{3} - \frac{1}{3} = \frac{2}{3}$$