Question

If $3^x=4^{x-1}$, then $x$ is equal to

• A. $\frac{2{\log }_{3}2}{2{\log }_{3}2–1}$
• B. $\frac{2}{2-{\log }_{2}3}$
• C. $\frac{1}{1-{\log }_{4}3}$
• D. $\frac{2{\log }_{2}3}{2{\log }_{2}3–1}$

Answer 1

The correct option is B

$\frac{2}{2-{\log }_{2}3}$

Explanation for correct option:
Given,
$$\begin{gathered} 3^x=4^{x-1} \\ \frac{3^x}{4^x}=\frac{1}{4} \end{gathered}$$

Apply logarithms of base 2 on both sides,

$\begin{array}{rcl}& \Rightarrow & x{\log }_{2}3–x{\log }_{2}4={\log }_{2}1–{\log }_{2}4\\ x({\log }_{2}3–2)& =& -2\\ x& =& \frac{2}{2-{\log }_{2}3}\end{array}$

Hence, the correct answer is option (B).