Questions: By taking natural logarithms of both sides of the equation, solve the equation 2 e^(3 x) = 11 Give your answer as a decimal number to three significant figures.

Solution Steps

To solve the equation \(2 e^{3x} = 11\), we can follow these steps:

1. Divide both sides of the equation by 2 to isolate the exponential term.
2. Take the natural logarithm (ln) of both sides to remove the exponential.
3. Solve for \(x\) by isolating it on one side of the equation.

Given the equation: \[ 2 e^{3x} = 11 \] Divide both sides by 2: \[ e^{3x} = \frac{11}{2} \]

Take the natural logarithm of both sides: \[ \ln(e^{3x}) = \ln\left(\frac{11}{2}\right) \]

Using the property \(\ln(e^y) = y\): \[ 3x = \ln\left(\frac{11}{2}\right) \]

Isolate \(x\) by dividing both sides by 3: \[ x = \frac{\ln\left(\frac{11}{2}\right)}{3} \]

Calculate the value to four significant digits: \[ x \approx 0.5682 \]

Final Answer