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

Using the property that log(a^b) = b*log(a), the equation can be simplified as follows:

2*log_2(x-1) - log_0.5(x-1) = 9

Now, we can combine the logarithms using the property that log_a(b) - log_a(c) = log_a(b/c):

log_2((x-1)^2 / log_0.5(x-1) = 9

Now, we can convert the equation to exponential form to solve for x.

2^9 = (x-1)^2 / 0.5^(x-1)

Simplifying, we get:

512 = (x-1)^2 / (1 / 0.5^(x-1))

Now, we can simplify the equation further:

512 = (x-1)^2 / (1 / 0.5^(x-1))
512 = (x-1)^2 * 2^(x-1)

At this point, we can use trial and error or a graphing calculator to find the value of x that satisfies this equation.