To solve the equation [tex]\( e^{5x} = 20 \)[/tex]:
1. Take the natural logarithm (ln) of both sides. The natural logarithm is the inverse operation of the exponential function, which allows us to deal with the exponent more easily.
[tex]\[
\ln(e^{5x}) = \ln(20)
\][/tex]
2. Apply the logarithm rule which states [tex]\( \ln(e^y) = y \)[/tex]. This simplifies our equation:
[tex]\[
5x = \ln(20)
\][/tex]
3. Isolate [tex]\( x \)[/tex] by dividing both sides of the equation by 5:
[tex]\[
x = \frac{\ln(20)}{5}
\][/tex]
4. Calculate the value.
The natural logarithm of 20 (approximately [tex]\( \ln(20) \approx 2.9957 \)[/tex]):
[tex]\[
x \approx \frac{2.9957}{5} \approx 0.6
\][/tex]
Therefore, the approximate solution to the equation [tex]\( e^{5x} = 20 \)[/tex] is:
Option A: [tex]\( x \approx 0.6 \)[/tex]
