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:
\(x \approx 2.047\).
