System of Nested Logarithms

CAT 2019 Slot 1 · Quantitative Ability · Easy · Logarithms

This is an easy Quantitative Ability question from the CAT 2019 Slot 1 paper. It tests Logarithms. 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 $\log_2(5 + \log_3 a) = 3$ and $\log_5(4a + 12 + \log_2 b) = 3$ , then $a + b$ is equal to:

A.
32
B.
59
C.
67
D.
40

Answer

Explanation

From first equation: $5 + \log_3 a = 2^3 = 8 \Rightarrow \log_3 a = 3 \Rightarrow a = 3^3 = 27$ .

Substitute $a = 27$ into second equation: $\log_5(4(27) + 12 + \log_2 b) = 3$ $\log_5(108 + 12 + \log_2 b) = 3 \Rightarrow 120 + \log_2 b = 5^3 = 125$ $\log_2 b = 5 \Rightarrow b = 2^5 = 32$ .

$a + b = 27 + 32 = 59$ .

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System of Nested Logarithms

Explanation

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