Introduction:
In this question, we are given the expression x^logy-logz . y^logz-logx . z^logx-logy and we need to evaluate it. We will break down the expression into individual parts and simplify each part step by step.

Expression Breakdown:
Let's evaluate each part of the given expression:

1. x^logy-logz:
- Using the property of logarithms, we can rewrite this as log(x^logy/logz).
- Applying the power rule of logarithms, we get (logy) * log(x) - log(z).

2. y^logz-logx:
- Similar to the previous part, we can rewrite this as log(y^logz/logx).
- Applying the power rule of logarithms, we get (logz) * log(y) - log(x).

3. z^logx-logy:
- Again, we can rewrite this part as log(z^logx/logy).
- Applying the power rule of logarithms, we get (logx) * log(z) - log(y).

Simplification:
Now, let's simplify the expression further by substituting the values obtained from the above steps:

1. x^logy-logz = (logy) * log(x) - log(z)
2. y^logz-logx = (logz) * log(y) - log(x)
3. z^logx-logy = (logx) * log(z) - log(y)

Final Evaluation:
To get the final evaluation, we substitute the simplified expressions back into the given expression:

x^logy-logz . y^logz-logx . z^logx-logy

= ((logy) * log(x) - log(z)) . ((logz) * log(y) - log(x)) . ((logx) * log(z) - log(y))

Conclusion:
In this question, we evaluated the expression x^logy-logz . y^logz-logx . z^logx-logy by breaking it down into individual parts and simplifying each part step by step. The final evaluation of the expression is ((logy) * log(x) - log(z)) . ((logz) * log(y) - log(x)) . ((logx) * log(z) - log(y)).