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: Let $\log_2 a = x$ and $\log_2 b = y$. Then $\log_2 a - \log_2 b = x - y = 2$.
Step 2: Since $ab=16$, $\log_2(ab) = \log_2 a + \log_2 b = x+y = \log_2 16 = 4$.
Step 3: Solve the two linear equations $x-y=2$ and $x+y=4$: adding gives $2x=6\Rightarrow x=3$; subtracting gives $2y=2\Rightarrow y=1$. So $\log_2 a = 3,\ \log_2 b = 1$.
Step 4: $\log_2(a^2 b^3) = 2x + 3y = 2(3) + 3(1) = 6+3=9$.
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