Question

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

Accepted Answer

C. $a = \log_5 2025 \Rightarrow \frac{1}{a} = \log_{2025} 5$. $b = \log_9 2025 \Rightarrow \frac{1}{b} = \log_{2025} 9$. \frac{1}{a} + \frac{1}{b} = \log_{2025} 5 + \log_{2025} 9 = \log_{2025} 45 = \log_{2025} \sqrt{2025} = \frac{1}{2}. Thus $\frac{a+b}{ab} = \frac{1}{2} \Rightarrow \frac{ab}{a+b} = 2$.

Logarithmic Relation with Exponential Terms

Explanation

Practise this under exam conditions