To solve the equation [tex]\( e^{2x} = 40 \)[/tex] and find the value of [tex]\( x \)[/tex], we can follow these steps:
1. Take the natural logarithm of both sides of the equation:
The natural logarithm (denoted as [tex]\(\ln\)[/tex]) is used because it is the inverse function of the exponential function with base [tex]\(e\)[/tex].
So, applying [tex]\(\ln\)[/tex] to both sides gives us:
[tex]\[
\ln(e^{2x}) = \ln(40)
\][/tex]
2. Simplify the left side using logarithm properties:
Recall that [tex]\(\ln(e^{y}) = y\)[/tex]. Applying this property:
[tex]\[
\ln(e^{2x}) = 2x
\][/tex]
Therefore, the equation simplifies to:
[tex]\[
2x = \ln(40)
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], divide both sides of the equation by 2:
[tex]\[
x = \frac{\ln(40)}{2}
\][/tex]
4. Calculate the value:
[tex]\(\ln(40)\)[/tex] is a constant value that can be approximated using a calculator. Substituting the natural logarithm of 40 in the equation and dividing by 2 results in:
[tex]\[
x \approx 1.844
\][/tex]
So, after rounding to the thousandths place, the value of [tex]\( x \)[/tex] is approximately [tex]\( 1.844 \)[/tex].
Comparing this value to the multiple-choice options provided, we see that:
- [tex]\( x = 0.801 \)[/tex]
- [tex]\( x = 3.689 \)[/tex]
- [tex]\( x = 1.602 \)[/tex]
- [tex]\( x = 1.844 \)[/tex]
The correct answer is [tex]\( \boxed{1.844} \)[/tex].
