A bacteria culture initially contains 100 cells and grows at a rate proportional to its size. After an hour the population has increased to 420.
a) Find an expression for the number of bacteria after t hours.
The formula I used is
P(t) = P(0) * e^(k*t)
I need P(0) --> We know that at t=0 P=100
So P(0) = 100
I need k
420 = 100 * e^(k*1)
100 = 100 * e^(k*0)
I took the ratio
420/100 = 100 * e^(k*1) / 100 = 100 * e^(k*0)
4.2 = e^k
ln 4.2 = ln e^k
ln 4.2 = k* lne
ln 4.2 = k* 1
Write the expression by plugging in P(0) and k
P(t) = 100 * e^[(ln4.2) *t]
The answer in the book is P(t) = 100 * (ln4.2)^t
I do not understand why
e^[(ln4.2) *t] = (ln4.2)^t
Can someone explain that to me please?