Question
If $2^x=3^y=12^z$, show that : $\frac{1}{z}=\frac{1}{y}+\frac{2}{x}$.
✓
Answer
Hint. Let $2^x=3^y=12^z=k$. Then, $2=k^{\frac{1}{x}}, 3=k^{\frac{1}{y}}$ and $12=k^{\frac{1}{x}}$.
Now, $12=2^2 \times 3 \Rightarrow k^{\frac{1}{x}}=\left(k^{\frac{1}{x}}\right)^2 \times\left(k^{\frac{1}{y}}\right) \Rightarrow k^{\frac{1}{x}}=k^{\left(\frac{2}{x}+\frac{1}{y}\right)}$
Now, $12=2^2 \times 3 \Rightarrow k^{\frac{1}{x}}=\left(k^{\frac{1}{x}}\right)^2 \times\left(k^{\frac{1}{y}}\right) \Rightarrow k^{\frac{1}{x}}=k^{\left(\frac{2}{x}+\frac{1}{y}\right)}$
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