To solve the equation [tex]\( e^{2x} = 40 \)[/tex], follow these steps:
1. Understand the equation: We have an exponential function where the base [tex]\( e \)[/tex] (Euler's number, approximately 2.718) is raised to the power of [tex]\( 2x \)[/tex] and set equal to 40.
2. Isolate [tex]\( 2x \)[/tex]: To isolate [tex]\( 2x \)[/tex], we need to get rid of the exponential. The most efficient way to do this is by taking the natural logarithm (ln) of both sides of the equation. This is because the natural logarithm and the exponential function are inverse operations.
[tex]\[
\ln(e^{2x}) = \ln(40)
\][/tex]
3. Simplify using properties of logarithms: The property of logarithms we need here is [tex]\( \ln(e^y) = y \)[/tex], thus:
[tex]\[
2x = \ln(40)
\][/tex]
4. Solve for [tex]\( x \)[/tex]: To solve for [tex]\( x \)[/tex], divide both sides of the equation by 2.
[tex]\[
x = \frac{\ln(40)}{2}
\][/tex]
5. Evaluate [tex]\( \ln(40) \)[/tex]: Using natural logarithm properties and calculations, we find that:
[tex]\[
\ln(40) \approx 3.689
\][/tex]
6. Complete the division:
[tex]\[
x \approx \frac{3.689}{2} \approx 1.844
\][/tex]
7. Round the result: According to the requirement to round to the thousandths place, we see that the result is [tex]\( x \approx 1.844 \)[/tex].
Therefore, the solution to the equation [tex]\( e^{2x} = 40 \)[/tex] is [tex]\( x \approx 1.844 \)[/tex].
From the provided options:
- [tex]\( x=0.801 \)[/tex]
- [tex]\( x=3.689 \)[/tex]
- [tex]\( x=1.602 \)[/tex]
- [tex]\( x=1.844 \)[/tex]
The correct rounded solution is:
[tex]\[ x = 1.844 \][/tex]
