CISE 8 Solve the following equations rounded off to three decimal places: \( \begin{array}{lll}\text { (a) } 3^{x}=4 & \text { (b) } 4^{2 x}=5 & \text { (c) } 3^{2 x+1}=12\end{array} \)

We are given the following equations to solve for x, rounding off to three decimal places: (a) 3ˣ = 4 (b) 4^(2x) = 5 (c) 3^(2x+1) = 12 We'll solve each one step by step. ──────────────────────────── Step 1. Equation (a): 3ˣ = 4 Take the natural logarithm (ln) on both sides:   ln(3ˣ) = ln(4) Use the logarithm property ln(aᵇ) = b ln(a) to get:   x · ln(3) = ln(4) Solve for x:   x = ln(4) / ln(3) Using a calculator:   ln(4) ≈ 1.386   ln(3) ≈ 1.099 Thus,   x ≈ 1.386 / 1.099 ≈ 1.262 ──────────────────────────── Step 2. Equation (b): 4^(2x) = 5 Take ln on both sides:   ln(4^(2x)) = ln(5) Apply the logarithm property:   2x · ln(4) = ln(5) Solve for x:   x = ln(5) / (2 ln(4)) Using approximate values:   ln(5) ≈ 1.609   ln(4) ≈ 1.386 Then,   x ≈ 1.609 / (2 × 1.386)   x ≈ 1.609 / 2.772 ≈ 0.580 ──────────────────────────── Step 3. Equation (c): 3^(2x+1) = 12 Take ln on both sides:   ln(3^(2x+1)) = ln(12) Apply the logarithm property:   (2x+1) · ln(3) = ln(12) Solve for x:   2x + 1 = ln(12) / ln(3) Subtract 1 from both sides:   2x = (ln(12) / ln(3)) - 1 Divide by 2:   x = [(ln(12) / ln(3)) - 1] / 2 Using approximate values:   ln(12) ≈ 2.485   ln(3) ≈ 1.099 Then,   x ≈ [(2.485 / 1.099) - 1] / 2   x ≈ (2.261 - 1) / 2   x ≈ 1.261 / 2 ≈ 0.631 ──────────────────────────── Rounded answers (to three decimal places): (a) x ≈ 1.262 (b) x ≈ 0.580 (c) x ≈ 0.631