Question

find the approximate solution to the following equation.
$e^{3x} = 2$
$x \approx \square$ (round to four decimal places as needed.)

Explanation:

Step1: Take natural log on both sides

To solve \( e^{3x} = 2 \), we take the natural logarithm (ln) of both sides. Recall that \( \ln(e^y)=y \) by the property of logarithms. So, applying \( \ln \) to both sides gives:
\( \ln(e^{3x})=\ln(2) \)
Using the logarithm property, the left - hand side simplifies to \( 3x \), so we have \( 3x=\ln(2) \)

Step2: Solve for x

Now, we solve the equation \( 3x = \ln(2) \) for \( x \) by dividing both sides by 3.
\( x=\frac{\ln(2)}{3} \)

Step3: Calculate the value

We know that \( \ln(2)\approx0.693147 \). Then, we substitute this value into the formula for \( x \):
\( x=\frac{0.693147}{3}\approx0.2310 \) (rounded to four decimal places)

Answer:

\( 0.2310 \)