Question: Evaluate the expression xlogy - logz ylogz - logx logx - logy

Solution:

To evaluate the expression xlogy - logz ylogz - logx logx - logy, we need to simplify each term separately and then combine the results.

Term 1: xlogy
To simplify xlogy, we can use the property of logarithms that states loga b^c = cloga b. Applying this property, we get:

xlogy = logy (x^logx)

Term 2: logz ylogz
To simplify logz ylogz, we can use the property of logarithms that states loga bc = loga b + loga c. Applying this property, we get:

logz ylogz = logz y + logz logz

Term 3: logx logx
The expression logx logx simplifies to (logx)^2.

Term 4: logy
The expression logy remains as it is.

Simplifying the terms:
Now that we have simplified each term separately, let's substitute the simplified terms back into the original expression and combine the results.

xlogy - logz ylogz - logx logx - logy
= logy (x^logx) - (logz y + logz logz) - (logx)^2 - logy

Combining like terms:
Next, let's combine the like terms together to simplify the expression further.

= logy (x^logx) - logz y - logz logz - (logx)^2 - logy

Final simplification:
The expression can be further simplified, but without any specific values or constraints given for the variables x, y, and z, we cannot simplify it any further. Therefore, the final expression is:

logy (x^logx) - logz y - logz logz - (logx)^2 - logy

In conclusion:
The expression xlogy - logz ylogz - logx logx - logy simplifies to logy (x^logx) - logz y - logz logz - (logx)^2 - logy. The expression can be further simplified if specific values or constraints are provided for the variables x, y, and z.