Question

If $x={\log }_{b}a$, $y={\log }_{c}b,$$z={\log }_{a}c$, then $xyz$ is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is B

$1$

Explanation for the correct answer:

By the property of logarithms,

${\log }_{m}n=\frac{\log n}{\log m}$

By using this property we can write

$\Rightarrow$$x={\log }_{b}a=\frac{\log a}{\log b}$$...\left(i\right)$

$\Rightarrow$$y={\log }_{c}b=\frac{\log b}{\log c}$$...\left(ii\right)$

$\Rightarrow$$z={\log }_{a}c=\frac{\log c}{\log a}$$...\left(iii\right)$

Multiplying $\left(i\right),\left(ii\right),\left(iii\right)$ we get

$\Rightarrow$$xyz=\frac{\log a}{\log b}\times \frac{\log b}{\log c}\times \frac{\log c}{\log a}$

$\Rightarrow$$xyz=1$

Thus the value of $xyz$ is $1$.

Hence, option (B) is the correct answer.