Basic Maths and Logarithm
Logarithmic Equations
JEE Advanced
Grade 11
Question:
If $x = 9$ is a solution of $\ln(x^2 + 15a^2) - \ln(a - 2) = \ln \frac{8ax}{a - 2}$, then which pair $(a, x)$ is also a solution?
(1) $(a = 3, x = 15)$
(2) $(a = 3/5, x = 2)$
(3) $(a = 9/5, x = 27/5)$
(4) $(a = 3, x = 2)$
Step-by-Step Solution
Key Concept: Simplify to $x^2 - 8ax + 15a^2 = 0$, i.e., $(x - 3a)(x - 5a) = 0$. With $x = 9$: $9 = 3a$ gives $a = 3$ (other root $x = 5a = 15$) or $9 = 5a$ gives $a = 9/5$ (other root $x = 3a = 27/5$). The pair $(a = 3, x = 15)$ is valid (check $a - 2 = 1 > 0$).
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)