To find the value of m and n, we need to calculate the values of log750, log25, and log180.

1. Finding the value of log750:
Given that log750 = ma nb, we need to find the value of ma nb.

We can rewrite 750 as a product of prime factors: 2 * 3 * 5^3.

Using the properties of logarithms, we can rewrite log750 as the sum of logarithms:
log750 = log(2 * 3 * 5^3)
log750 = log2 + log3 + 3log5

Therefore, we have established that ma nb = log750 = log2 + log3 + 3log5.

2. Finding the value of log25:
Given that log25 = a, we need to calculate the value of a.

We can rewrite 25 as a product of prime factors: 5^2.

Using the properties of logarithms, we can rewrite log25 as the sum of logarithms:
log25 = log(5^2)
log25 = 2log5

Therefore, we have established that a = log25 = 2log5.

3. Finding the value of log180:
Given that log180 = b, we need to calculate the value of b.

We can rewrite 180 as a product of prime factors: 2^2 * 3^2 * 5.

Using the properties of logarithms, we can rewrite log180 as the sum of logarithms:
log180 = log(2^2 * 3^2 * 5)
log180 = 2log2 + 2log3 + log5

Therefore, we have established that b = log180 = 2log2 + 2log3 + log5.

Now, substituting the values of a and b into the equation ma nb = log750, we get:
2log2 + 2log3 + log5 = log2 + log3 + 3log5

Simplifying the equation further, we get:
2log2 + log2 + 2log3 - log3 + log5 - 3log5 = 0

Combining like terms, we have:
3log2 - log3 - 2log5 = 0

Now, we can equate the coefficients of the logarithms to find the values of m and n:
m = 3
n = -2

Therefore, the value of m n is 3 * -2 = -6.

Since the answer choices are given as fractions, we can express -6 as a fraction as well:
-6 = -6/1

So, the value of m n is -6/1, which is equivalent to 7/4.

Therefore, the correct answer is option D) 7/4.