To solve this logarithmic equation, we will first use the properties of logarithms to rewrite the equation in a simpler form.

Apply the power rule of logarithms, which states that loga(b^c) = c*loga(b):

log^36(2x-7)^2 = log^6(14-x)

Apply the rule of logarithms that states if loga(b) = loga(c), then b = c:

(2x-7)^2 = 14-x

Square the left side of the equation:

4x^2 - 28x + 49 = 14 - x

Rearrange the terms:

4x^2 - 27x + 35 = 0

Factor the quadratic equation:

(4x - 5)(x - 7) = 0

Set each factor to zero and solve for x:

4x - 5 = 0 -> 4x = 5 -> x = 5/4
x - 7 = 0 -> x = 7

Therefore, the solutions to the equation are x = 5/4 and x = 7.