Basic Maths and Logarithm
Exponential and Logarithmic Equations
JEE Main 2023
Grade 11
Question:
Let $S = \{\alpha : \log_2(9^{2\alpha - 4} + 13) - \log_2(\frac{5}{2} \cdot 3^{2\alpha - 4} + 1) = 2\}$. The number of elements in $S$ is:
(1) $0$
(2) $1$
(3) $2$
(4) $\infty$
Step-by-Step Solution
Key Concept: Let $t = 3^{2\alpha - 4} > 0$: equation becomes $(t^2 + 13)/(5t/2 + 1) = 4 \Rightarrow t^2 - 10t + 9 = 0 \Rightarrow t = 1$ or $t = 9$. $t = 1 \Rightarrow \alpha = 2$; $t = 9 \Rightarrow \alpha = 3$. Both valid, so $|S| = 2$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (3)