1. The problem is to solve the equation $|2^x| = 2^9$. 2. Since $2^x$ is always positive for any real $x$, the absolute value does not change the expression. So, $|2^x| = 2^x$. 3. Therefore, the equation simplifies to: $$2^x = 2^9$$ 4. When the bases are the same and positive (and not equal to 1), the exponents must be equal: $$x = 9$$ 5. Hence, the solution to the equation is $x = 9$.