4.2. \( \frac{1}{\sqrt{2}}=4^{x-1} \)

To tackle the equation \( \frac{1}{\sqrt{2}}=4^{x-1} \), we can start by rewriting \( 4 \) as \( 2^2 \). Therefore, we can express \( 4^{x-1} \) as \( (2^2)^{x-1} = 2^{2(x-1)} \). Now our equation looks like: \[ \frac{1}{\sqrt{2}} = 2^{2(x-1)} \] Next, we recognize that \( \frac{1}{\sqrt{2}} = 2^{-1/2} \). This lets us rewrite the equation as: \[ 2^{-1/2} = 2^{2(x-1)} \] Since the bases are the same, we can set the exponents equal to each other: \[ -\frac{1}{2} = 2(x - 1) \] Now, solving for \( x \) involves simplifying further: 1. Distributing the \( 2 \): \[ -\frac{1}{2} = 2x - 2 \] 2. Adding \( 2 \) to both sides: \[ \frac{3}{2} = 2x \] 3. Dividing by \( 2 \) gives: \[ x = \frac{3}{4} \] Thus, the solution to the equation is \( x = \frac{3}{4} \).