Rewrite the expression using logarithm properties.

The expression to evaluate is $\log_2 8 \sqrt{\frac{a^5 b}{c^3}}$.
Using the product rule of logarithms, $\log_x (MN) = \log_x M + \log_x N$:

Rewrite the square root as a power: $\sqrt{X} = X^{\frac{1}{2}}$.

Using the power rule, $\log_x M^P = P \log_x M$:

Using the quotient rule, $\log_x \left(\frac{M}{N}\right) = \log_x M - \log_x N$:

Using the product rule, $\log_x (MN) = \log_x M + \log_x N$:

Since $2^3 = 8$, we have $\log_2 8 = 3$.

Given:
$\log_2 a = 3$
$\log_2 b = -2$
$\log_2 c = 5$
Substitute these values into the expression from Step 4: