Start with the equation \(4^{x-2} = 4^x - 4^2\).

Rewrite \(4^{x-2}\) as \(\frac{4^x}{4^2}\), so the equation becomes:

\(\frac{4^x}{16} = 4^x - 16\).

Multiply through by 16 to clear the fraction:

\(4^x = 16(4^x - 16)\).

Expand the right side:

\(4^x = 16 \cdot 4^x - 256\).

Rearrange to isolate terms involving \(4^x\):

\(4^x - 16 \cdot 4^x = -256\).

Factor out \(4^x\):

\(4^x(1 - 16) = -256\).

Simplify:

\(-15 \cdot 4^x = -256\).

Divide both sides by -15:

\(4^x = \frac{256}{15}\).

Take the logarithm of both sides to solve for \(x\):

\(x \log(4) = \log\left(\frac{256}{15}\right)\).

Solve for \(x\):

\(x = \frac{\log\left(\frac{256}{15}\right)}{\log(4)}\).

Calculate the value:

\(x \approx 2.047\).