Proving an identity of logarithms

Question

Prove that $$\log_xy\log_zx\log_yz = 1$$

I have tried changing the base but I can never get the same base for all the logs so that it can possibly cancel each other out to get $1.$

Answer 1

In fact $x=X$, $y=Y$ and $z=Z$.

Using $\log_e(.)=\log(.)$

$$\log_x(y)=\frac{\log (y)}{\log (x)}\qquad \log_z(x)=\frac{\log (x)}{\log (z)}\qquad \log_y(z)=\frac{\log (z)}{\log (y)}$$