To solve the equation [tex]\( 7e^{6x} = 42 \)[/tex], we can follow these steps:
1. Isolate the exponential term:
[tex]\[
e^{6x} = \frac{42}{7}
\][/tex]
Simplifying the right-hand side, we get:
[tex]\[
e^{6x} = 6
\][/tex]
2. Take the natural logarithm of both sides:
Taking the natural logarithm (ln) on both sides helps to deal with the exponential:
[tex]\[
\ln(e^{6x}) = \ln(6)
\][/tex]
3. Simplify using logarithm properties:
Use the property [tex]\(\ln(e^{k}) = k \cdot \ln(e)\)[/tex] where [tex]\( \ln(e) = 1 \)[/tex]:
[tex]\[
6x \cdot \ln(e) = \ln(6)
\][/tex]
Simplifying further:
[tex]\[
6x = \ln(6)
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides by 6:
[tex]\[
x = \frac{\ln(6)}{6}
\][/tex]
The numerical approximation for [tex]\( x \)[/tex] using the given answer results in:
[tex]\[
x \approx 0.30
\][/tex]
Thus, the correct answer is:
[tex]\[
x \approx 0.30
\][/tex]
