If 3x = 4x–1, then x =

Engineering

Mathematics

Properties of Logarithm

Question

If 3^x = 4^x–1, then x =

$\frac{2 \log_{2} 3}{2 \log_{2} 3 - 1}$

$\frac{2 \log_{3} 2}{2 \log_{3} 2 - 1}$

$\frac{2}{2 - \log_{2} 3}$

$\frac{1}{1 - \log_{4} 3}$

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3^x = 4^{x – 1}

⇒ log₃^3x = log₃4^{(x – 1)} ⇒ x = (x – 1)log₃4 ⇒ x = 2(x – 1) log₃2

⇒ 2 log₃2 = x(2 log₃2 – 1)

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