CAT 2022 Slot 2 QA Q17 Logarithms & Exponents

Question 17 of the Quantitative Ability (QA) section, CAT 2022 Slot 2, with CAT's official answer and a worked solution.

The number of integer solutions of the equation \(\left(x^2-10\right)^{\left(x^2-3 x-10\right)}=1\) is

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

Answer: 4 Logarithms & Exponents TITA

Solution

Exponent 0: \(x^2-3x-10=0\Rightarrow x=5,-2\); the bases 15 and \(-6\) are nonzero, so both work.
Base 1: \(x^2=11\), no integers.
Base \(-1\) with an even exponent: \(x=\pm3\); exponents \(-10\) and 8 are even, so both work.
Total 4 solutions.