CAT 2025 Slot 3 QA Q7 Logarithms & Exponents

Question 7 of the Quantitative Ability (QA) section, CAT 2025 Slot 3, with CAT's official answer and a worked solution.

The sum of all possible real values of \( x \) for which \( \log _ { x - 3 } \left( x ^ { 2 } - 9 \right) = \log _ { x - 3 } ( x + 1 ) + 2 \), is

A
\( - 3 \)
B
\( \sqrt { 33 } \)
C
\( \frac { 3 + \sqrt { 33 } } { 2 } \)
D
\( 3 \)

Answer: (C) ( 3 + √( 33 ) )/( 2 ) Logarithms & Exponents MCQ

Solution

The base needs \(x - 3 > 0\), \(x - 3 \ne 1\): so \(x > 3\), \(x \ne 4\).
Write \(2 = \log_{x-3}(x-3)^2\): then \(x^2 - 9 = (x + 1)(x - 3)^2\).
Divide by \(x - 3\): \(x + 3 = (x + 1)(x - 3) = x^2 - 2x - 3 \Rightarrow x^2 - 3x - 6 = 0\).
\(x = \frac{3 \pm \sqrt{33}}{2}\); only \(\frac{3 + \sqrt{33}}{2} \approx 4.37\) is valid, so the sum is \(\frac{3 + \sqrt{33}}{2}\).