Questions: Solve the exponential equation algebraically. Approximate the result to three decimal places. 2 e^x = 18

Solution Steps

To solve the exponential equation \(2 e^{x} = 18\), we first isolate the exponential term by dividing both sides by 2, resulting in \(e^{x} = 9\). Then, we take the natural logarithm of both sides to solve for \(x\), which gives us \(x = \ln(9)\). Finally, we approximate the result to three decimal places.

Starting with the equation: \[ 2 e^{x} = 18 \] we divide both sides by 2 to isolate the exponential term: \[ e^{x} = 9 \]

Next, we take the natural logarithm of both sides: \[ \ln(e^{x}) = \ln(9) \] Using the property of logarithms, this simplifies to: \[ x = \ln(9) \]

Calculating the natural logarithm, we find: \[ x \approx 2.1972245773362196 \] Rounding this to three decimal places gives: \[ x \approx 2.197 \]

Final Answer