Question

Convert the given exponential function to the form indicated. Round all coefficients to four significant digits. f(t) = 2.8(1.001)^t; f(t) Qoe^kt

Answer 1

you know that \(Q_0=2.8\) because when you replace \(t\) by \(0\) you get \(2.8\)

Answer 2

now if \(t=1\) the left hand side is \(2.8(1.001)=2.8028\) so you can set
\[2.8028=2.8e^k\] and solve for \(k\)

Answer 3

in two steps
a) divide by \(2.8\)
b) take the log of both sides to find \(k\)

Answer 4

I dont get what I take the log of exactly?

Answer 5

\[2.8028 = 2.8 e^k\]
\[\frac{2.8028}{2.8} = e^k\]
Now take the natural log of both sides:
\[\ln{\frac{2.8028}{2.8}} = \ln{e^k}\]But the natural log of e^x is just x, so
\[k = \ln{2.8028/2.8} = \ln{1.001} \approx 0.000995\]

Answer 6

Sorry, I formatted the last line poorly. It should be
\[k = \ln{\frac{2.8028}{2.8}} = \ln{1.001} \approx 0.000995\]

Answer 7

My eyes are going, the value approximately 0.0009995...