CAT 2021 Set-2 | Quantitative Aptitude | Question: 19

Question

If $\log_{2} [3+ \log_{3} \{ 4+ \log_{4} (x-1) \}] – 2 = 0$ then $4x$ equals

Answer 1

Given that, $\log_{2}\left[3+\log_{3}\{4+\log_{4}(x-1)\}\right]-2=0$

$\Rightarrow \log_{2}\left[3+\log_{3}\{4+\log_{4}(x-1)\}\right]=2$

$\Rightarrow 3+\log_{3}\{4+\log_{4}(x-1)\}=2^{2}\quad \left[\because \log_{a}{b} = x \Rightarrow b=a^{x}\right]$

$\Rightarrow \log_{3}\{4+\log_{4}(x-1)\}=1$

$\Rightarrow 4+\log_{4}(x-1)=3^{1}$

$\Rightarrow \log_{4}(x-1)=-1$

$\Rightarrow x-1=4^{-1}$

$\Rightarrow x= \frac{1}{4}+1$

$\Rightarrow \boxed{x= \frac{5}{4}}$

$\therefore$ The value of $4x = 4 \times \frac{5}{4} = 5.$

Correct Answer $:5$