Explanation:
Given, xy = x^x
Taking log on both sides, we get,
log(xy) = log(x^x)
Using the property of logarithms, we can simplify the equation as follows:
log(x) + log(y) = x log(x)

Now, we are required to find the value of 1/(log x(y) 1)

Solution:

Step 1: Simplifying the expression
log x(y) 1 can be written as log x(1/y)
So, 1/(log x(y) 1) = 1/(log x(1/y))

Using the property of logarithms, we can simplify this expression as follows:
1/(log x(1/y)) = 1/(log x - log y)

Step 2: Substituting the values
From the equation we derived initially, we have:
log(x) + log(y) = x log(x)

Rearranging the terms, we get:
log(x) - x log(x) = -log(y)

Dividing both sides by log(x), we get:
1 - x = -log(y)/log(x)

Substituting this value in the expression for 1/(log x(y) 1), we get:
1/(log x(y) 1) = -log(x)/[log(y) - x log(x)]

Therefore, 1/(log x(y) 1) = -log(x)/[log(y) - x log(x)] is the final solution.