Question

If $\frac{\log x}{b-c}=\frac{\log y}{c-a}=\frac{\log z}{a-b}$, then which of the following is/are true?

• A. $xyz=1$
• B. $x^a{y}^b{z}^c=1$
• C. $x^{b+c}{y}^{c+a}{z}^{a+b}=1$
• D. $xyz=x^a{y}^b{z}^c$

Answer 1

Answer: (A), (B), (C), (D)

$xyz=x^a{y}^b{z}^c$

$\frac{{\log }_{k}x}{b-c}=\frac{{\log }_{k}y}{c-a}=\frac{{\log }_{k}z}{a-b}=p\text{(let)}\Rightarrow \frac{{\log }_{k}x}{b-c}=p\Rightarrow x=k^{\left(p\right(b-c\left)\right)}\text{Similarly,}y=k^{p(c-a)}\text{and}z=k^{p(a-b)}xyz=k^{p(b-c+c-a+a-b)}=1x^a{y}^b{z}^c=k^{p(ab-ac+bc-ab+ac-bc)}=1x^{b+c}{y}^{c+a}{z}^{a+b}=k^{p(b^2-c^2+c^2-a^2+a^2-b^2)}=1$