CAT 2025 Slot 3 QA Q9 Logarithms & Exponents

Question 9 of the Quantitative Ability (QA) section, CAT 2025 Slot 3, with CAT's official answer and a worked solution.

If \( 12 ^ { 12 x } \times 4 ^ { 24 x + 12 } \times 5 ^ { 2 y } = 8 ^ { 4 z } \times 20 ^ { 12 x } \times 243 ^ { 3 x - 6 } \), where \( x , y \) and \( z \) are natural numbers, then \( x + y + z \) equals

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

Answer: 112 Logarithms & Exponents TITA

Solution

LHS \(= 2^{24x} 3^{12x} \cdot 2^{48x + 24} \cdot 5^{2y} = 2^{72x + 24} 3^{12x} 5^{2y}\).
RHS \(= 2^{12z} \cdot 2^{24x} 5^{12x} \cdot 3^{15x - 30}\).
Powers of 3: \(12x = 15x - 30 \Rightarrow x = 10\). Powers of 5: \(2y = 120 \Rightarrow y = 60\).
Powers of 2: \(744 = 12z + 240 \Rightarrow z = 42\). So \(x + y + z = 112\).