To solve this logarithmic equation, we will first start by simplifying the logarithmic terms.

Using the properties of logarithms:

4*log9(x-2) + log3(x-4)^2 = 0
log9(x-2)^4 + log3(x-4)^2 = 0

Now we can combine the logarithms using the property that states log(a) + log(b) = log(a*b):

log9[(x-2)^4 * x^2-8x+16] = 0

Now we can convert the equation into exponential form:

9^0 = (x-2)^4 * (x^2 - 8x + 16)

Since any non-zero number raised to the power of 0 is 1, we have:

1 = (x-2)^4 * (x^2 - 8x + 16)

Expanding the terms on the right side of the equation:

1 = (x-2)(x-2)(x-2)(x-2) * (x-4)(x-4)

1 = (x-2)^4 * (x-4)^2

Now, in order for the equation to be true, the only solution for x would be when:

(x-2)^4 * (x-4)^2 = 1

Since 1 is a constant value and both terms on the right side must multiply to equal 1, we can conclude that:

(x-2)^4 = 1 and (x-4)^2 = 1

Taking the fourth root of both sides of the first equation results in:

(x-2) = 1 or (x-2) = -1

Solving for x in these equations, we find two possible solutions:

x = 3 or x = 1

Therefore, the solutions to the given logarithmic equation are x = 3 and x = 1.