To solve the equation [tex]\( e^{-0.314 t} = 7 \)[/tex] using natural logarithms, follow these steps:
1. Take the natural logarithm of both sides:
This helps isolate the exponential part of the equation.
[tex]\[
\ln(e^{-0.314 t}) = \ln(7)
\][/tex]
2. Use the property of logarithms:
Recall that [tex]\(\ln(e^x) = x\)[/tex]. This allows us to simplify the left side of the equation.
[tex]\[
-0.314 t = \ln(7)
\][/tex]
3. Solve for [tex]\( t \)[/tex]:
To isolate [tex]\( t \)[/tex], divide both sides by [tex]\(-0.314\)[/tex].
[tex]\[
t = \frac{\ln(7)}{-0.314}
\][/tex]
4. Calculate the value:
Evaluating [tex]\(\ln(7)\)[/tex] gives us approximately 1.945910. Now, divide this by [tex]\(-0.314\)[/tex]:
[tex]\[
t \approx \frac{1.945910}{-0.314} \approx -6.197
\][/tex]
So, the solution set is [tex]\( \{ -6.197 \} \)[/tex].
