We need to show that:

$$(\log_x y) (\log_z x) (\log_y z) = 1$$

Step 1: Expressing Logarithms in Terms of Natural Logarithms

Using the change of base formula:

$$\log_a b = \frac{\ln b}{\ln a}$$

we rewrite each term:

$$\log_x y = \frac{\ln y}{\ln x}, \quad \log_z x = \frac{\ln x}{\ln z}, \quad \log_y z = \frac{\ln z}{\ln y}$$

Step 2: Multiply the Three Expressions

$$\left( \frac{\ln y}{\ln x} \right) \times \left( \frac{\ln x}{\ln z} \right) \times \left( \frac{\ln z}{\ln y} \right)$$

Step 3: Simplification

Since the terms in the numerators and denominators cancel out:

$$\frac{\ln y}{\ln x} \times \frac{\ln x}{\ln z} \times \frac{\ln z}{\ln y} = 1$$

Thus, we have:

$$(\log_x y)(\log_z x)(\log_y z) = 1$$

Conclusion

The given logarithmic identity is proven to be true.