What is the solution to the equation [tex]\(e^{3x}=12\)[/tex]? Round your answer to the nearest hundredth.

A. [tex]\(x = 0.83\)[/tex]
B. [tex]\(x = 1.09\)[/tex]
C. [tex]\(x = 2.48\)[/tex]
D. [tex]\(x = 7.44\)[/tex]



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]