Given equation:
X^2 Y^2 = 14 XY

To find: 2 log 4 log x log y

Solution:

1. Simplifying the equation:
X^2 Y^2 = 14 XY

We can rewrite this equation as:
X^2 Y^2 - 14 XY = 0

Factoring out the common terms, we get:
XY (X^2 Y - 14) = 0

2. Solving for XY:
From the equation, we have two possible solutions:
1) XY = 0
2) X^2 Y - 14 = 0

2.1. Solving for XY = 0:
If XY = 0, it means either X = 0 or Y = 0.

2.1.1. X = 0:
If X = 0, then log X is undefined.

2.1.2. Y = 0:
If Y = 0, then log Y is undefined.

2.2. Solving for X^2 Y - 14 = 0:
X^2 Y - 14 = 0
X^2 Y = 14

Taking the logarithm of both sides, we get:
log (X^2 Y) = log (14)

Using the logarithmic property, we can separate the terms:
log X^2 + log Y = log 14

3. Finding 2 log 4 log X log Y:
To find 2 log 4 log X log Y, we need to find the values of log X and log Y.

3.1. Finding log X:
From the equation log X^2 + log Y = log 14, we can isolate log X:
log X^2 = log 14 - log Y
log X^2 = log (14/Y)

Applying the exponent rule, we have:
2 log X = log (14/Y)

Dividing both sides by 2, we get:
log X = log (14/Y) / 2

3.2. Finding log Y:
From the equation log X^2 + log Y = log 14, we can isolate log Y:
log Y = log 14 - log X^2
log Y = log 14 - 2 log X

4. Calculating 2 log 4 log X log Y:
Using the values of log X and log Y obtained above, we can calculate 2 log 4 log X log Y:

2 log 4 log X log Y = 2 log 4 (log (14/Y) / 2) (log 14 - 2 log X)

5. Conclusion:
The value of 2 log 4 log X log Y cannot be determined without knowing the specific values of X and Y.