If 3 x = 4 x - 1 , then x = [2013]

Question

If $3^{x} = 4^{x - 1},$ then $x =$ [2013]

Smart Achievers · Accepted Answer

(1, 2, 3)

$3^{x} = 4^{x - 1} \Rightarrow x \log 3 = 2 (x - 1) \log 2$

$\Rightarrow x = \frac{2 \log 2}{2 \log 2 - \log 3}$ $\Rightarrow x = \frac{2 \log_{3} 2}{2 \log_{3} 2 - 1} = \frac{2}{2 - \log_{2} 3}$

$Also x = \frac{1}{1 - \frac{1}{2} \log_{2} 3} = \frac{1}{1 - \log_{4} 3}$

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