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Logarithmic equation solving

CAT 2021 Slot 2 · Quantitative Ability · Medium · Logarithms

This is a medium Quantitative Ability question from the CAT 2021 Slot 2 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[3+log⁡3{4+log⁡4(x−1)}]−2=0\log_2 [3 + \log_3 \{4 + \log_4 (x - 1)\}] - 2 = 0, then 4x4x equals

Answer

5

Explanation

Given: log⁡2[3+log⁡3{4+log⁡4(x−1)}]=2\log_2 [3 + \log_3 \{4 + \log_4 (x - 1)\}] = 2 3+log⁡3{4+log⁡4(x−1)}=22=43 + \log_3 \{4 + \log_4 (x - 1)\} = 2^2 = 4 log⁡3{4+log⁡4(x−1)}=1\log_3 \{4 + \log_4 (x - 1)\} = 1 4+log⁡4(x−1)=31=34 + \log_4 (x - 1) = 3^1 = 3 log⁡4(x−1)=−1\log_4 (x - 1) = -1 x−1=4−1=14x - 1 = 4^{-1} = \frac{1}{4} x=1+14=54x = 1 + \frac{1}{4} = \frac{5}{4}

Therefore, 4x=4×54=54x = 4 \times \frac{5}{4} = 5.

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