Question

If $a^x=b,b^y=c,c^z=a$, show that $xyz=1$.

Answer 1

Given: $a^x=b,b^y=c,c^z=a$

To prove: $xyz=1$

Taking $\log$ for the equations, we have

$x\log a=\log b$ ...(i)

$y\log b=\log c$ ...(ii)

$z\log c=\log a$ ... (iii)

Now, $\left(1\right)\times \left(2\right)\times \left(3\right)$

$\left(xyz\right)\log a\log b\log c=\log a\log b\log c$

$\Rightarrow xyz=1$