Sure, let's solve the equation [tex]\( e^{0.4x} = 0.4 \)[/tex] step by step.
1. Understanding the Problem:
We need to solve for [tex]\( x \)[/tex] in the equation [tex]\( e^{0.4x} = 0.4 \)[/tex].
2. Taking Natural Logarithm:
To solve for [tex]\( x \)[/tex], we can take the natural logarithm (ln) of both sides of the equation. The natural logarithm has the property that [tex]\( \ln(e^y) = y \)[/tex].
[tex]\[
\ln(e^{0.4x}) = \ln(0.4)
\][/tex]
3. Simplifying using Logarithm Properties:
Using the property [tex]\( \ln(e^y) = y \)[/tex], we simplify the left-hand side of the equation:
[tex]\[
0.4x = \ln(0.4)
\][/tex]
4. Solving for [tex]\( x \)[/tex]:
Now we need to isolate [tex]\( x \)[/tex] by dividing both sides of the equation by 0.4:
[tex]\[
x = \frac{\ln(0.4)}{0.4}
\][/tex]
5. Calculating the Natural Logarithm and Division:
Evaluate [tex]\( \ln(0.4) \approx -0.916290731874155 \)[/tex]. Then perform the division:
[tex]\[
x = \frac{-0.916290731874155}{0.4} \approx -2.2907268296853873
\][/tex]
6. Rounding the Result:
Finally, round the result to two decimal places:
[tex]\[
x \approx -2.29
\][/tex]
Based on the above calculations and rounding, the solution to the equation [tex]\( e^{0.4x} = 0.4 \)[/tex] is approximately [tex]\( x = -2.29 \)[/tex].
So, the correct answer is:
D. [tex]\( x = -2.29 \)[/tex]
