Introduction:
In this problem, we are given two equations involving logarithms of two variables x and y. We are required to find the value of another logarithmic expression in terms of x and y.

Solution:

We can start by simplifying the given equations:

log x = m n => x = e^(m n) [exponentiating both sides with base e]

log y = m - n => y = e^(m - n)

Now, we need to find the value of log (10x/y^2).

Let's first simplify this expression:

log (10x/y^2) = log 10 + log x - 2 log y [using logarithmic identities]

= 1 + log x - 2(log y) [log 10 = 1]

= 1 + log (e^(m n)) - 2(log(e^(m - n))) [substituting the values of x and y]

= 1 + mn - 2(m - n)

= 3n - mn + 1

Therefore, the value of log (10x/y^2) is 3n - mn + 1.

Conclusion:
In this problem, we used the given equations to simplify the expression log (10x/y^2) in terms of m and n. The final answer is 3n - mn + 1.