Answer :
To solve the equation [tex]\( e^{3x} = 12 \)[/tex] for [tex]\( x \)[/tex], you can follow these steps:
1. Take the natural logarithm of both sides of the equation:
[tex]\[ \ln(e^{3x}) = \ln(12) \][/tex]
2. Simplify the left side using the property of logarithms that states [tex]\( \ln(e^y) = y \)[/tex]:
[tex]\[ 3x = \ln(12) \][/tex]
3. Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 3:
[tex]\[ x = \frac{\ln(12)}{3} \][/tex]
4. Calculate the natural logarithm of 12. For our purposes, we already know the numerical value is approximately:
[tex]\[ \ln(12) \approx 2.4849066498 \][/tex]
5. Divide this value by 3:
[tex]\[ x \approx \frac{2.4849066498}{3} \][/tex]
6. Perform the division to find [tex]\( x \)[/tex]:
[tex]\[ x \approx 0.8283022166 \][/tex]
7. Round the result to the nearest hundredth:
[tex]\[ x \approx 0.83 \][/tex]
Therefore, the solution to the equation [tex]\( e^{3x} = 12 \)[/tex] rounded to the nearest hundredth is [tex]\( x = 0.83 \)[/tex]. So, the correct answer from the provided options is:
[tex]\[ x = 0.83 \][/tex]
1. Take the natural logarithm of both sides of the equation:
[tex]\[ \ln(e^{3x}) = \ln(12) \][/tex]
2. Simplify the left side using the property of logarithms that states [tex]\( \ln(e^y) = y \)[/tex]:
[tex]\[ 3x = \ln(12) \][/tex]
3. Solve for [tex]\( x \)[/tex] by dividing both sides of the equation by 3:
[tex]\[ x = \frac{\ln(12)}{3} \][/tex]
4. Calculate the natural logarithm of 12. For our purposes, we already know the numerical value is approximately:
[tex]\[ \ln(12) \approx 2.4849066498 \][/tex]
5. Divide this value by 3:
[tex]\[ x \approx \frac{2.4849066498}{3} \][/tex]
6. Perform the division to find [tex]\( x \)[/tex]:
[tex]\[ x \approx 0.8283022166 \][/tex]
7. Round the result to the nearest hundredth:
[tex]\[ x \approx 0.83 \][/tex]
Therefore, the solution to the equation [tex]\( e^{3x} = 12 \)[/tex] rounded to the nearest hundredth is [tex]\( x = 0.83 \)[/tex]. So, the correct answer from the provided options is:
[tex]\[ x = 0.83 \][/tex]
