2 log 2 [ x / 3 ] = 2 + log 2 ( x − 8 ) .

To solve this logarithmic equation, we first need to rewrite it in exponential form.

The given equation is:

2log2(x/3) = 2 + log2(x-8)

Using the properties of logarithms, we can rewrite this as:

log2((x/3)^2) = log2(2) + log2(x-8)

Now, we can combine the logarithms on the right side using the product rule:

log2((x/3)^2) = log2(2(x-8))

Next, we can eliminate the logarithms by setting the expressions inside the logarithms equal to each other:

(x/3)^2 = 2(x-8)

Now, we can simplify and solve for x:

(x^2/9) = 2x - 16

Multiply both sides by 9 to get rid of the fraction:

x^2 = 18x - 144

Rearrange the equation to set it equal to zero:

x^2 - 18x + 144 = 0

Now, we can factor the quadratic equation:

(x - 12)(x - 12) = 0

So, the solution to the equation is x = 12.