Logarithmic Equation Solution

CAT 2023 Slot 1 · Quantitative Ability · Easy · Algebra

This is an easy Quantitative Ability question from the CAT 2023 Slot 1 paper. It tests Algebra. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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If $x$ and $y$ are positive real numbers such that $\log_x (x^2 + 12) = 4$ and $3 \log_y x = 1$ , then $x+y$ equals

A.
20
B.
11
C.
68
D.
10

Answer

Explanation

Given $\log_x (x^2 + 12) = 4 \implies x^4 = x^2 + 12$ . Let $x^2 = t$ . Then $t^2 - t - 12 = 0 \implies (t - 4)(t + 3) = 0$ . Since $t = x^2 > 0$ , we have $t = 4 \implies x = 2$ (since $x > 0$ ).

Also, $3 \log_y x = 1 \implies \log_y x = \frac{1}{3} \implies y^{1/3} = 2 \implies y = 8$ .

Therefore, $x + y = 2 + 8 = 10$ .

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Logarithmic Equation Solution

Explanation

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