Answer :
To solve the equation [tex]\(5 e^{5x} = 20\)[/tex], let's proceed with a detailed, step-by-step process.
1. Isolate the exponential expression:
[tex]\[ 5 e^{5x} = 20 \][/tex]
Divide both sides by 5 to isolate [tex]\(e^{5x}\)[/tex]:
[tex]\[ e^{5x} = \frac{20}{5} = 4 \][/tex]
2. Take the natural logarithm of both sides:
Apply the natural logarithm ([tex]\(\ln\)[/tex]) to both sides of the equation to get rid of the exponent:
[tex]\[ \ln(e^{5x}) = \ln(4) \][/tex]
3. Simplify using logarithm properties:
Use the property of logarithms that [tex]\(\ln(e^y) = y\)[/tex]:
[tex]\[ 5x = \ln(4) \][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 5 to isolate [tex]\(x\)[/tex]:
[tex]\[ x = \frac{\ln(4)}{5} \][/tex]
5. Approximate the value of [tex]\(x\)[/tex]:
Using a calculator or known logarithm values, we find that [tex]\(\ln(4) \approx 1.386294361\)[/tex]. Thus:
[tex]\[ x \approx \frac{1.386294361}{5} \approx 0.277258872 \][/tex]
Given this result, the approximate solution to the equation [tex]\(5 e^{5 x} = 20\)[/tex] is:
A. [tex]\(x \approx 0.28\)[/tex]
So, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
1. Isolate the exponential expression:
[tex]\[ 5 e^{5x} = 20 \][/tex]
Divide both sides by 5 to isolate [tex]\(e^{5x}\)[/tex]:
[tex]\[ e^{5x} = \frac{20}{5} = 4 \][/tex]
2. Take the natural logarithm of both sides:
Apply the natural logarithm ([tex]\(\ln\)[/tex]) to both sides of the equation to get rid of the exponent:
[tex]\[ \ln(e^{5x}) = \ln(4) \][/tex]
3. Simplify using logarithm properties:
Use the property of logarithms that [tex]\(\ln(e^y) = y\)[/tex]:
[tex]\[ 5x = \ln(4) \][/tex]
4. Solve for [tex]\(x\)[/tex]:
Divide both sides by 5 to isolate [tex]\(x\)[/tex]:
[tex]\[ x = \frac{\ln(4)}{5} \][/tex]
5. Approximate the value of [tex]\(x\)[/tex]:
Using a calculator or known logarithm values, we find that [tex]\(\ln(4) \approx 1.386294361\)[/tex]. Thus:
[tex]\[ x \approx \frac{1.386294361}{5} \approx 0.277258872 \][/tex]
Given this result, the approximate solution to the equation [tex]\(5 e^{5 x} = 20\)[/tex] is:
A. [tex]\(x \approx 0.28\)[/tex]
So, the correct answer is:
[tex]\[ \boxed{A} \][/tex]
