Basic Maths and Logarithm
Logarithmic Equations
JEE Main 2021
Grade 11

Question:

The number of solutions of $\log_4(x - 1) = \log_2(x - 3)$ is:
(1) $0$
(2) $1$
(3) $2$
(4) $3$

Step-by-Step Solution

Key Concept: Rewrite LHS: $\frac{1}{2} \log_2(x - 1) = \log_2(x - 3) \Rightarrow \log_2(x - 1) = 2 \log_2(x - 3) \Rightarrow x - 1 = (x - 3)^2$. Solve: $x^2 - 7x + 10 = 0 \Rightarrow x = 5$ or $x = 2$. Domain: $x > 3$ (for $\log_2(x - 3)$). Only $x = 5$ valid.
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Correct Answer: (2)

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