Question

If \( ab = 16 \) and \( \log_2 a - \log_2 b = 2 \), find the value of \( \log_2 a^2 b^3 \).

• A. 8
• B. 9
• C. 10
• D. 11
• E. 12

Hint:

When dealing with logarithms, remember the logarithmic properties: \( \log_b x^n = n \log_b x \) and \( \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y \).

Answer 1

The Correct Option is B

Step 1: Analyzing the given equations.
We are given \( ab = 16 \) and \( \log_2 a - \log_2 b = 2 \). We can use logarithmic properties to simplify these expressions. From the second equation, apply the property \( \log_2 a - \log_2 b = \log_2 \left( \frac{a}{b} \right) \). Thus, we have: \[ \log_2 \left( \frac{a}{b} \right) = 2 \] This implies: \[ \frac{a}{b} = 2^2 = 4 \] Therefore, we can express \( a \) as: \[ a = 4b \] Step 2: Substituting into the first equation.
Now substitute \( a = 4b \) into the first equation \( ab = 16 \): \[ (4b)b = 16 \] \[ 4b^2 = 16 \] \[ b^2 = 4 \] \[ b = 2 \] Now substitute \( b = 2 \) into \( a = 4b \): \[ a = 4 \times 2 = 8 \] Step 3: Finding the value of \( \log_2 a^2 b^3 \).
Now, we need to find \( \log_2 a^2 b^3 \). We can simplify this as: \[ \log_2 a^2 b^3 = \log_2 a^2 + \log_2 b^3 \] Using the properties of logarithms, we can rewrite it as: \[ \log_2 a^2 + \log_2 b^3 = 2 \log_2 a + 3 \log_2 b \] Substitute \( a = 8 \) and \( b = 2 \) into this expression: \[ 2 \log_2 8 + 3 \log_2 2 \] We know that \( \log_2 8 = 3 \) and \( \log_2 2 = 1 \), so: \[ 2 \times 3 + 3 \times 1 = 6 + 3 = 9 \] Final Answer: \[ \boxed{9} \]