To solve the equation [tex]\( 5e^{5x} = 20 \)[/tex], follow these steps:
1. Isolate the exponential term: Start by dividing both sides of the equation by 5 to isolate the exponential expression.
[tex]\[
e^{5x} = \frac{20}{5} = 4
\][/tex]
2. Apply the natural logarithm: To solve for [tex]\( x \)[/tex], take the natural logarithm (denoted as [tex]\(\ln\)[/tex]) of both sides of the equation.
[tex]\[
\ln(e^{5x}) = \ln(4)
\][/tex]
3. Simplify using logarithm properties: The natural logarithm of an exponential function simplifies as follows:
[tex]\[
5x = \ln(4)
\][/tex]
4. Solve for [tex]\( x \)[/tex]: Finally, solve for [tex]\( x \)[/tex] by dividing both sides by 5.
[tex]\[
x = \frac{\ln(4)}{5}
\][/tex]
The value of [tex]\(\ln(4)\)[/tex] is approximately [tex]\(1.386\)[/tex]. To find [tex]\( x \)[/tex]:
[tex]\[
x \approx \frac{1.386}{5} \approx 0.277
\][/tex]
Now, we compare this calculated value with the given options to find the closest approximation:
- A. [tex]\( x \approx 0.40 \)[/tex]
- B. [tex]\( x \approx 0.28 \)[/tex]
- C. [tex]\( x \approx 0.22 \)[/tex]
- D. [tex]\( x \approx 1.86 \)[/tex]
Seeing that [tex]\(\approx 0.277\)[/tex] is closest to [tex]\( 0.28 \)[/tex].
Therefore, the correct answer is:
B. [tex]\( x \approx 0.28 \)[/tex]
