3) \( \lg ^{2} x-3 \lg x=4 \)

To solve the equation \( \lg^2 x - 3 \lg x - 4 = 0 \), we can start by making a substitution. Let \( y = \lg x \). This changes our equation to \( y^2 - 3y - 4 = 0 \). Now, we can factor this quadratic equation: \[ (y - 4)(y + 1) = 0 \] Setting each factor to zero gives us: \[ y - 4 = 0 \quad \Rightarrow \quad y = 4 \] \[ y + 1 = 0 \quad \Rightarrow \quad y = -1 \] Now substituting back for \( y \): 1. For \( y = 4 \): \[ \lg x = 4 \quad \Rightarrow \quad x = 10^4 = 10000 \] 2. For \( y = -1 \): \[ \lg x = -1 \quad \Rightarrow \quad x = 10^{-1} = 0.1 \] Thus, the solutions to the equation \( \lg^2 x - 3 \lg x - 4 = 0 \) are \( x = 10000 \) and \( x = 0.1 \).