Answer :

To solve the equation [tex]\( 7000 = 100e^{-0.03t} \)[/tex] for [tex]\( t \)[/tex], follow these steps:

1. Isolate the exponential term: Divide both sides of the equation by 100 to simplify:
[tex]\[
70 = e^{-0.03t}
\][/tex]

2. Apply the natural logarithm: Take the natural logarithm (ln) of both sides to eliminate the exponential function:
[tex]\[
\ln(70) = \ln(e^{-0.03t})
\][/tex]

3. Simplify the equation using logarithm properties: The property [tex]\( \ln(e^x) = x \)[/tex] allows us to simplify the right side:
[tex]\[
\ln(70) = -0.03t
\][/tex]

4. Solve for [tex]\( t \)[/tex]: Multiply both sides by [tex]\(-\frac{1}{0.03}\)[/tex] (which is approximately [tex]\(-33.33\)[/tex]) to isolate [tex]\( t \)[/tex]:
[tex]\[
t = -\frac{\ln(70)}{0.03}
\][/tex]

5. Calculate the numerical value: Compute the value of the expression to find the exact value of [tex]\( t \)[/tex]:
[tex]\[
t \approx -141.616508068312
\][/tex]

There are also complex solutions due to the periodic nature of certain functions involved, but the primary solution here is the real value [tex]\( t \approx -141.62 \)[/tex].

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Rewritten by : Barada