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NDA 2019 Maths · Mathematics · Logarithms · easy

If log ab=2 and log bc=3, then log ac equals

Single correct option

If log⁡ab=2\log_ab=2 and log⁡bc=3\log_bc=3, then log⁡ac\log_ac equals
  1. A. 66
  2. B. 55
  3. C. 23\dfrac23
  4. D. 32\dfrac32
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Answer: (A) 66

By the chain rule for logarithms, log⁡ac=log⁡ab⋅log⁡bc=2×3=6\log_ac=\log_ab\cdot\log_bc=2\times3=6. This follows from the change of base formula: log⁡blog⁡a⋅log⁡clog⁡b=log⁡clog⁡a\dfrac{\log b}{\log a}\cdot\dfrac{\log c}{\log b}=\dfrac{\log c}{\log a}. The option 55 adds instead of multiplying, which would be the rule for products inside a logarithm, not for logarithms of different bases.
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Original question written in the NDA 2019 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake