To solve this equation, we can take the logarithm of both sides with respect to a common base, say 2. This gives:
8^x = 2^(3x) and 3^(x+2) = 2^(log2(3)*(x+2))
So our equation becomes:
2^(3x) = 2^(log2(3)*(x+2))
Now we can equate the exponents on both sides:
3x = log2(3)*(x+2)
Solving for x, we get:
3x = log2(3)*x + 2*log2(3)
2x = 2*log2(3)
x = log2(3)
Therefore, the solution to the equation 8^x = 3^x+2 is x = log2(3).