To solve the equation [tex]\( e^{3x} = 12 \)[/tex], follow these steps:
1. Identify the equation:
The given equation is [tex]\( e^{3x} = 12 \)[/tex].
2. Take the natural logarithm (ln) of both sides:
Using the properties of logarithms, specifically the natural logarithm which is the inverse of the exponential function with base [tex]\( e \)[/tex], we can take the natural logarithm of both sides of the equation:
[tex]\[
\ln(e^{3x}) = \ln(12)
\][/tex]
3. Simplify the left side using the logarithm property [tex]\( \ln(e^y) = y \)[/tex]:
[tex]\[
3x = \ln(12)
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], divide both sides of the equation by 3:
[tex]\[
x = \frac{\ln(12)}{3}
\][/tex]
5. Calculate [tex]\( \ln(12) \)[/tex]:
The natural logarithm of 12 is approximately 2.48490664979.
6. Divide by 3:
[tex]\[
x = \frac{2.48490664979}{3} \approx 0.8283022166
\][/tex]
7. Round the result to the nearest hundredth:
[tex]\[
x \approx 0.83
\][/tex]
8. Confirm the correct option:
Among the provided options:
[tex]\[
\boxed{x = 0.83}
\][/tex]
is the value that matches our rounded solution.
Thus, the solution to the equation [tex]\( e^{3x} = 12 \)[/tex], rounded to the nearest hundredth, is [tex]\( x = 0.83 \)[/tex].
