To solve this logarithmic equation, we need to use the properties of logarithms.

First, we can simplify the equation by using the property that log(a) - log(b) = log(a/b). Applying this to the equation, we get:

log7((x-2)/(x+2)) = log7(7/(2x-7))

Since the bases of the logarithms are the same, we can drop the logs on both sides and set the contents of each side equal to each other:

(x-2)/(x+2) = 7/(2x-7)

Next, we can cross multiply to get rid of the fractions:

(x-2)(2x-7) = 7(x+2)

Expanding both sides:

2x^2 - 7x - 4x + 14 = 7x + 14

2x^2 - 11x + 14 = 7x + 14

2x^2 - 11x - 7x = 0

2x^2 - 18x = 0

2x(x - 9) = 0

So, we have two possible solutions: x = 0 or x = 9.

However, we need to check if either of these solutions satisfies the original equation. Plugging in x = 0 into the original equation, we get:

log7(-2) - log7(2) = 1 - log7(-7)

This results in a non-integer output, so x = 0 is not a solution.

Plugging in x = 9 into the original equation, we get:

log7(7) - log7(11) = 1 - log7(13)

This equation holds true, so x = 9 is the solution to the given logarithmic equation.