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CAT 2021 Slot 2 — QA Question 11

LogarithmsEasy

log2 [3 + log3 {4 + log4 (x - 1)}] - 2 = 0, then 4x equals

Answer & solution

Answer: 5

Solution

Easy

Unwrap the nested logarithm from the outside in. Each time, use log⁡bM=k⇒M=bk\log_b M=k \Rightarrow M=b^{k} to peel off one layer until only xx remains.

1

Isolate the outer log. The equation is log⁡2 ⁣[3+log⁡3{4+log⁡4(x−1)}]−2=0\log_2\!\big[3+\log_3\{4+\log_4(x-1)\}\big]-2=0.

log⁡2 ⁣[3+log⁡3{4+log⁡4(x−1)}]=2⇒ 3+log⁡3{4+log⁡4(x−1)}=22=4\begin{aligned} &\log_2\!\big[3+\log_3\{4+\log_4(x-1)\}\big]=2\\ &\Rightarrow\ 3+\log_3\{4+\log_4(x-1)\}=2^{2}=4 \end{aligned}
2

Peel the middle log.

log⁡3{4+log⁡4(x−1)}=1⇒ 4+log⁡4(x−1)=31=3\begin{aligned} &\log_3\{4+\log_4(x-1)\}=1\\ &\Rightarrow\ 4+\log_4(x-1)=3^{1}=3 \end{aligned}
3

Peel the inner log and solve for xx.

log⁡4(x−1)=−1⇒ x−1=4−1=14⇒ x=54\begin{aligned} &\log_4(x-1)=-1\\ &\Rightarrow\ x-1=4^{-1}=\tfrac{1}{4}\\ &\Rightarrow\ x=\tfrac{5}{4} \end{aligned}
4

Compute 4x4x.

4x=4×54=5\begin{aligned} &4x=4\times\tfrac{5}{4}=5 \end{aligned}
4x=54x=5
CAT 2021 Slot 2 QA Q11: log 2 [3 + log 3 {4 + log 4 (x - 1)}] - 2 = 0, then 4x equals — Solution | TheCATExam