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Logarithmic Relation with Exponential Terms

CMAT 2023 · Quantitative Techniques & Data Interpretation · Easy · Logarithms

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If 5a=9b=20255^a = 9^b = 2025, then the value of aba+b=\frac{ab}{a+b} = _____.

  1. A.

    4

  2. B.

    3

  3. C.

    2

  4. D.

    1

Answer

C

Explanation

a=log⁡52025⇒1a=log⁡20255a = \log_5 2025 \Rightarrow \frac{1}{a} = \log_{2025} 5. b=log⁡92025⇒1b=log⁡20259b = \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 a+bab=12⇒aba+b=2\frac{a+b}{ab} = \frac{1}{2} \Rightarrow \frac{ab}{a+b} = 2.

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