We can use the properties of logarithms to simplify the equation:

Use the properties of logarithms: log7(2x^2-7x+6) - log7(x-2) = log7(x)
log7 ((2x^2-7x+6)/(x-2)) = log7(x)

Since the bases are the same, the expressions inside the logs must be equal:
(2x^2-7x+6)/(x-2) = x

Multiply both sides by x-2 to get rid of the fraction:
2x^2 - 7x + 6 = x(x - 2)
2x^2 - 7x + 6 = x^2 - 2x

Rearrange the equation to set it equal to zero:
x^2 - 5x + 6 = 0

Factor the quadratic equation:
(x-2)(x-3) = 0

The solutions for x are x = 2 and x = 3.

Therefore, the solutions for the original equation are x = 2 and x = 3.