CAT 2024 Slot 1 QA Q22 Logarithms & Exponents

Question 22 of the Quantitative Ability (QA) section, CAT 2024 Slot 1, with CAT's official answer and a worked solution.

If \(x\) is a positive real number such that \(4 \log _{10} x+4 \log _{100} x+8 \log _{1000} x=13\), then the greatest integer not exceeding \(x\), is

TITA: type in the answer. There are no options and no negative marking.

Answer: 31 Logarithms & Exponents TITA

Solution

  1. Let \(L = \log_{10} x\). Then \(\log_{100} x = \frac{L}{2}\) and \(\log_{1000} x = \frac{L}{3}\).
  2. \(4L + 2L + \frac{8L}{3} = 13 \Rightarrow \frac{26L}{3} = 13 \Rightarrow L = \frac32\).
  3. \(x = 10^{1.5} = 10\sqrt{10} \approx 31.62\); the greatest integer not exceeding \(x\) is \(31\).

Same concept in other years

See every question on Logs with bases that are powers of one base: convert to a common base.

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