Basic Maths and Logarithm
Logarithmic Equations
JEE Advanced 2011 Paper 1
Grade 11
Question:
Let $(x_0, y_0)$ be the solution of the system $(2x)^{\ln 2} = (3y)^{\ln 3}$ and $3^{\ln x} = 2^{\ln y}$. Then $x_0$ is:
(1) $\frac{1}{6}$
(2) $\frac{1}{3}$
(3) $\frac{1}{2}$
(4) $6$
Step-by-Step Solution
Key Concept: Take $\ln$ of both equations. Let $p = \ln x, q = \ln y, a = \ln 2, b = \ln 3$. From the second equation: $pb = qa \Rightarrow q = pb/a$. Substitute into the first to get $a(a + p) = b(b + q)$; solve to find $p = -a$, so $x = e^{-\ln 2} = 1/2$.
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Correct Answer: (3)