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JEE Advanced 2013 · Linear Inequalities

If 3x=4x−13^{x} = 4^{x-1}, then x=x =

Answers: A. 2log⁡322log⁡32−1\frac{2\log_3 2}{2\log_3 2 - 1} · B. 22−log⁡23\frac{2}{2 - \log_2 3} · C. 11−log⁡43\frac{1}{1 - \log_4 3}

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Written solution

If 3^{x} = 4^{x-1}, then x = 3x=4x−13^{x} = 4^{x-1} Find x\text{Find } x Take log⁡3 on both sides\text{Take } \log_3 \text{ on both sides} x=(x−1)log⁡34x = (x-1)\log_3 4 log⁡34=log⁡322=2log⁡32\log_3 4 = \log_3 2^2 = 2\log_3 2 x=(x−1) (2log⁡32)x = (x-1)\,(2\log_3 2) x=2xlog⁡32−2log⁡32x = 2x\log_3 2 - 2\log_3 2 x (1−2log⁡32)=−2log⁡32x\,(1 - 2\log_3 2) = -2\log_3 2 x=−2log⁡321−2log⁡32=2log⁡322log⁡32−1x = \dfrac{-2\log_3 2}{1 - 2\log_3 2} = \dfrac{2\log_3 2}{2\log_3 2 - 1} log⁡32≈0.63 ⇒ x≈4.82\log_3 2 \approx 0.63 \ \Rightarrow \ x \approx 4.82 x=2log⁡322log⁡32−1x = \dfrac{2\log_3 2}{2\log_3 2 - 1} Answer: Option (1)\boxed{\text{Answer: Option (1)}}

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