To solve for [tex]\( x \)[/tex] in the equation [tex]\( 2 e^{3x} = 400 \)[/tex], follow these steps:
1. Isolate the exponential term:
[tex]\[
e^{3x} = \frac{400}{2} = 200
\][/tex]
2. Take the natural logarithm of both sides:
[tex]\[
\ln(e^{3x}) = \ln(200)
\][/tex]
Using the property of logarithms that [tex]\( \ln(e^y) = y \)[/tex], we get:
[tex]\[
3x = \ln(200)
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{\ln(200)}{3}
\][/tex]
4. Using a calculator, find the natural logarithm of 200:
[tex]\[
\ln(200) \approx 5.298
\][/tex]
5. Divide by 3 to solve for [tex]\( x \)[/tex]:
[tex]\[
x \approx \frac{5.298}{3} \approx 1.766
\][/tex]
Therefore, the solution for [tex]\( x \)[/tex] is approximately [tex]\( 1.766 \)[/tex]. Hence, the answer is:
D) 1.766
