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

Answer 1

The Correct Option is A

Step 1: Use the properties of logarithms:

• \(\log_a b = \log a - \log b\) 
• \(\log a^n = n \log a\)

Step 2: Simplify the expression:

\[ \log \frac{75}{16} - 2 \log \frac{5}{9} + \log \frac{32}{243} \]

Expanding using logarithm properties:

\[ \log 75 - \log 16 - 2(\log 5 - \log 9) + \log 32 - \log 243 \]

Step 3: Simplify further:

\[ \log 75 - \log 16 - 2\log 5 + 2\log 9 + \log 32 - \log 243 \]

Using logarithm multiplication property:

\[ \log \frac{75 \times 81 \times 32}{16 \times 25 \times 243} \]

Final result:

\[ \log 2 \]