Solve for the variable. $$ \log _{8} 2-\log _{8} 5 x=2 $$ .0063 25.6 9.32

Question

UpStudy AI · Accepted Answer

Rewrite the equation using the logarithm subtraction rule:

$$
\log_{8}2 - \log_{8}(5x) = \log_{8}\left(\frac{2}{5x}\right)
$$

So the equation becomes:

$$
\log_{8}\left(\frac{2}{5x}\right) = 2
$$

Convert the logarithmic equation to its exponential form:

$$
\frac{2}{5x} = 8^2
$$

Since $$8^2 = 64$$, we have:

$$
\frac{2}{5x} = 64
$$

Solve for $$x$$ by first isolating $$5x$$:

$$
5x = \frac{2}{64}
$$

Simplify the right-hand side:

$$
5x = \frac{1}{32}
$$

Now divide both sides by $$5$$:

$$
x = \frac{1}{32 \cdot 5} = \frac{1}{160}
$$

The decimal approximation of $$\frac{1}{160}$$ is:

$$
x \approx 0.00625
$$

Among the given choices, the option that best matches is:

$$
\boxed{0.0063}
$$
$$ x \approx 0.0063 $$

Solve for the variable.

.0063
25.6
9.32