Question

If \(5^a\) = \(9^b\) = 2025, then the value of \(\frac{ab}{a+b}\)= ____.

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

The Correct Option is C

 We are given the equation \(5^a = 9^b = 2025\), and we need to find the value of \(\frac{ab}{a+b}\).

1. First, express 2025 in its prime factors: \(2025 = 5^2 \times 9^2 = (5 \times 9)^2 = 45^2\)
2. Since \(5^a = 2025\), we can equate the powers: \(5^a = 45^2\)\)
3. Similarly, for \(9^b = 2025\): \(9^b = 45^2\)\)
4. Now, compute \(\frac{ab}{a+b}\): 
   Substitute the expressions for \(a\) and \(b\): \(\frac{ab}{a+b} = \frac{(2 \log_5{45})(2 \log_9{45})}{2 \log_5{45} + 2 \log_9{45}}\)
   Simplifying: \(= \frac{4 \log_5{45} \log_9{45}}{2(\log_5{45} + \log_9{45})}\)
   \(= 2 \cdot \frac{\log_5{45} \log_9{45}}{\log_5{45} + \log_9{45}}\)
5. Note that the expression simplifies to: \(\frac{\log_5{45} \log_9{45}}{\log_5{45} + \log_9{45}} = 1\)
   Therefore: \(\frac{ab}{a+b} = 2 \times 1 = 2\)

Thus, the correct answer is 2.