Step 1: 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, $\log_x (MN) = \log_x M + \log_x N$: $$ \log_2 8 + \log_2 \sqrt{\frac{a^5 b}{c^3}} $$ Rewrite the square root as a power: $\sqrt{X} = X^{\frac{1}{2}}$. $$ \log_2 8 + \log_2 \left(\frac{a^5 b}{c^3}\right)^{\frac{1}{2}} $$ Step 2: Apply the power rule of logarithms. Using the power rule, $\log_x M^P = P \log_x M$: $$ \log_2 8 + \frac{1}{2} \log_2 \left(\frac{a^5 b}{c^3}\right) $$ Step 3: Apply the quotient and product rules of logarithms inside the parenthesis. Using the quotient rule, $\log_x \left(\frac{M}{N}\right) = \log_x M - \log_x N$: $$ \log_2 8 + \frac{1}{2} (\log_2 (a^5 b) - \log_2 (c^3)) $$ Using the product rule, $\log_x (MN) = \log_x M + \log_x N$: $$ \log_2 8 + \frac{1}{2} (\log_2 a^5 + \log_2 b - \log_2 c^3) $$ Step 4: Apply the power rule again for $a^5$ and $c^3$. $$ \log_2 8 + \frac{1}{2} (5 \log_2 a + \log_2 b - 3 \log_2 c) $$ Step 5: Evaluate $\log_2 8$. Since $2^3 = 8$, we have $\log_2 8 = 3$. Step 6: Substitute the given values into the expression. Given: $\log_2 a = 3$ $\log_2 b = -2$ $\log_2 c = 5$ Substitute these values into the expression from Step 4: $$ 3 + \frac{1}{2} (5(3) + (-2) - 3(5)) $$ Step 7: Perform the arithmetic calculations. $$ 3 + \frac{1}{2} (15 - 2 - 15) $$ $$ 3 + \frac{1}{2} (-2) $$ $$ 3 - 1 $$ $$ 2 $$ The final answer is $\boxed{2}$.