Answer :

To solve the equation [tex]\( 8603 = 7000 e^{6.4 t} \)[/tex] for [tex]\( t \)[/tex], follow these steps:

1. Isolate the Exponential Expression:
Begin by dividing both sides of the equation by 7000 to isolate the exponential term. The equation becomes:
[tex]\[
\frac{8603}{7000} = e^{6.4 t}
\][/tex]

2. Simplify the Fraction:
Calculate the fraction on the left side:
[tex]\[
\frac{8603}{7000} \approx 1.229
\][/tex]
So, the equation now is:
[tex]\[
1.229 = e^{6.4 t}
\][/tex]

3. Use the Natural Logarithm:
Take the natural logarithm (ln) of both sides to solve for [tex]\( t \)[/tex]:
[tex]\[
\ln(1.229) = \ln(e^{6.4 t})
\][/tex]

4. Apply the Power Rule of Logarithms:
Using the property [tex]\(\ln(e^x) = x\)[/tex], the equation simplifies to:
[tex]\[
\ln(1.229) = 6.4 t
\][/tex]

5. Solve for [tex]\( t \)[/tex]:
Divide both sides by 6.4 to find [tex]\( t \)[/tex]:
[tex]\[
t = \frac{\ln(1.229)}{6.4}
\][/tex]

6. Calculate the Value of [tex]\( t \)[/tex]:
By computing the value, we find:
[tex]\[
t \approx 0.032
\][/tex]

So, the solution for [tex]\( t \)[/tex] is approximately [tex]\( 0.032 \)[/tex].

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