Question

find the approximate solution to the following equation.
$e^{5x} = 7$
$x \approx \square$ (round to four decimal places as needed.)

Explanation:

Step1: Take natural log on both sides

To solve \( e^{5x}=7 \), 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:
\( \ln(e^{5x})=\ln(7) \)

Step2: Simplify left side

Using the logarithm property \( \ln(e^y)=y \), the left side simplifies to \( 5x \). So we have:
\( 5x = \ln(7) \)

Step3: Solve for x

Divide both sides by 5 to isolate \( x \):
\( x = \frac{\ln(7)}{5} \)

Step4: Calculate the value

We know that \( \ln(7) \approx 1.945910149 \). Dividing this by 5:
\( x \approx \frac{1.945910149}{5} \approx 0.3891820298 \)

Step5: Round to four decimals

Rounding \( 0.3891820298 \) to four decimal places gives \( 0.3892 \).

Answer:

\( x \approx \boxed{0.3892} \)