Question

c. $12 = 3e^{-x + 5}$

Explanation:

Step1: Divide both sides by 3

Divide each side of the equation \(12 = 3e^{-x + 5}\) by 3. This gives \(\frac{12}{3}=\frac{3e^{-x + 5}}{3}\), which simplifies to \(4=e^{-x + 5}\).

Step2: Take natural logarithm of both sides

Take the natural logarithm (ln) of both sides. Recall that \(\ln(e^{a})=a\). So, \(\ln(4)=\ln(e^{-x + 5})\), which simplifies to \(\ln(4)=-x + 5\).

Step3: Solve for x

First, subtract 5 from both sides: \(\ln(4)-5=-x\). Then, multiply both sides by - 1 to solve for x: \(x = 5-\ln(4)\). We can also write \(\ln(4)\) as \(2\ln(2)\), so \(x = 5 - 2\ln(2)\approx5 - 2\times0.693=5 - 1.386 = 3.614\).

Answer:

\(x = 5-\ln(4)\) (or \(x = 5 - 2\ln(2)\) or the approximate decimal value \(x\approx3.614\))