Search any question & find its solution
Question:
Answered & Verified by Expert
If $2^{x}=3^{y}=12^{2}$, then what is $(x+2 y) /(x y)$ equal to ?
Options:
Solution:
2517 Upvotes
Verified Answer
The correct answer is:
$\frac{1}{z}$
Given that $2^{\mathrm{x}}=3^{\mathrm{y}}=12^{\mathrm{z}}=\mathrm{k}$
Taking $\log _{2}$ on both the sides $x=\log _{2} k, y=\log _{3} k$ and $z=\log _{12} k$
$\frac{x+2 y}{x y}=\frac{\log _{2} k+2 \log _{3} k}{\log _{2} k \log _{3} k}$
$=\frac{1}{\log _{3} \mathrm{k}}+\frac{2}{\log _{2} \mathrm{k}}$
$=\log _{\mathrm{k}} 3+2 \log _{\mathrm{k}} 2=\log _{\mathrm{k}} 3+\log _{\mathrm{k}} 4$
$=\log _{\mathrm{k}} 12=\frac{1}{\log _{12} \mathrm{k}}=\frac{1}{\mathrm{z}}$
Taking $\log _{2}$ on both the sides $x=\log _{2} k, y=\log _{3} k$ and $z=\log _{12} k$
$\frac{x+2 y}{x y}=\frac{\log _{2} k+2 \log _{3} k}{\log _{2} k \log _{3} k}$
$=\frac{1}{\log _{3} \mathrm{k}}+\frac{2}{\log _{2} \mathrm{k}}$
$=\log _{\mathrm{k}} 3+2 \log _{\mathrm{k}} 2=\log _{\mathrm{k}} 3+\log _{\mathrm{k}} 4$
$=\log _{\mathrm{k}} 12=\frac{1}{\log _{12} \mathrm{k}}=\frac{1}{\mathrm{z}}$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.