The problem asks us to solve several exponential equations. Let's solve the first equation (i) $2^{2x} - 5(2^x) + 4 = 0$.

1. Problem Description

The problem asks us to solve several exponential equations. Let's solve the first equation (i) $2^{2x} - 5(2^x) + 4 = 0$.

2. Solution Steps

(i) $2^{2x} - 5(2^x) + 4 = 0$

Let $u = 2^x$. Then the equation becomes $u^2 - 5u + 4 = 0$.

We can factor the quadratic as $(u - 1)(u - 4) = 0$.

Therefore, $u = 1$ or $u = 4$.

If $u = 1$, then $2^x = 1 = 2^0$, so $x = 0$.

If $u = 4$, then $2^x = 4 = 2^2$, so $x = 2$.

3. Final Answer

$x = 0, 2$