CAT 2024 Slot 1 QA Q3 Logarithms & Exponents

Question 3 of the Quantitative Ability (QA) section, CAT 2024 Slot 1, with CAT's official answer and a worked solution.

The sum of all real values of \(k\) for which \(\left(\frac{1}{8}\right)^k \times\left(\frac{1}{32768}\right)^{\frac{1}{3}}=\frac{1}{8} \times\left(\frac{1}{32768}\right)^{\frac{1}{k}}\), is

A
\(\frac{2}{3}\)
B
\(-\frac{2}{3}\)
C
\(\frac{4}{3}\)
D
\(-\frac{4}{3}\)

Answer: (B) -2/3 Logarithms & Exponents MCQ

Solution

Use base 2: \(\frac18 = 2^{-3}\) and \(32768 = 2^{15}\), so \(\left(\frac{1}{32768}\right)^{1/3} = 2^{-5}\).
The equation becomes \(2^{-3k} \cdot 2^{-5} = 2^{-3} \cdot 2^{-15/k}\), so \(-3k - 5 = -3 - \frac{15}{k}\).
Multiply by \(k\): \(-3k^2 - 2k + 15 = 0\), i.e. \(3k^2 + 2k - 15 = 0\).
The discriminant \(4 + 180 > 0\), so both roots are real. Their sum \(= -\frac{2}{3}\).