Question

Simplify: \( \log \frac{75}{16} - 2 \log \frac{5}{9} + \log \frac{32}{243} \)

• A. \( \log 2 \)
• B. \( \log 4 \)
• C. \( \log 1 \)
• D. \( \log 3 \)
• E.

Hint:

Use logarithmic properties to combine and simplify the terms: \[ \log \frac{a}{b} = \log a - \log b \quad \text{and} \quad \log a^n = n \log a. \]

Answer 1

The Correct Option is A

Simplify the given logarithmic expression:
log₁₆75 − 2 log₅9 + log₃₂243

Step 1: Convert logarithms to base 2 for simplification.
log₁₆75 = log₂75 / log₂16 = log₂75 / 4
log₅9 = log₂9 / log₂5 = log₂9 / log₂(5²) = log₂9 / (2 log₂5)
log₃₂243 = log₂243 / log₂32 = log₂243 / 5

Step 2: Apply transformations.
2 log₅9 = 2 (log₂9 / 2 log₂5) = log₂9 / log₂5
Rewriting the expression:
(log₂75 / 4) - (log₂9 / log₂5) + (log₂243 / 5)

Step 3: Approximate values and simplify further (if needed).
Final simplified form: log₂(75^1/4 * 243^1/5 / 9^1/log₂5)