Question

If $\log _3 5=x$ and $\log _{25} 11=y$ then the value of $\log _3\left(\frac{11}{3}\right)$ in terms of $x$ and $y$ is

• A. $x^2 y-1$
• B. $2 x y-1$
• C. $2 x y+1$
• D. $\frac{2 y-x}{x}$

Answer 1

Answer: (B)

$ \log _3 5= x \Rightarrow 3^{ x }=5 \Rightarrow 5^{1 / x }=3 $ ....(1)
$\text { and } \log _{25} 11=\frac{\log _5 11}{2}= y$
$\therefore \log _5 11=2 y \Rightarrow 5^{2 y }=11 $....(2)
$(2 \div 1) \Rightarrow \frac{11}{3}=5^{2 y-\frac{1}{x}}=5^{\frac{2 x y-1}{x}} $
$\log _3\left(\frac{11}{3}\right)=\frac{2 x y-1}{x} \log _3 5=\frac{2 x y-1}{x} \cdot x=2 x y-1 .$