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NDA 2023 Maths · Mathematics · Logarithms · medium

Log 10x<0 if and only if 0<x<1. 2. log ab=-log ba for all positive a,b different from 1.

Single correct option

Consider the following statements:
1. log⁡10x<0\log_{10}x<0 if and only if 0<x<10<x<1.
2. log⁡ab=−log⁡ba\log_ab=-\log_ba for all positive a,ba,b different from 11.
Which of the statements given above is/are correct?
  1. A. 1 only
  2. B. 2 only
  3. C. Both 1 and 2
  4. D. Neither 1 nor 2
Show answer and solution

Answer: (A) 1 only

Statement 1 is true because log⁡10\log_{10} is increasing, log⁡101=0\log_{10}1=0, and the function is only defined for x>0x>0. Statement 2 is false: log⁡ab⋅log⁡ba=1\log_ab\cdot\log_ba=1, so log⁡ab=1log⁡ba\log_ab=\dfrac1{\log_ba}, the reciprocal and not the negative. For example log⁡24=2\log_24=2 but log⁡42=12\log_42=\tfrac12.
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Original question written in the NDA 2023 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