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**DBA** 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)

Step 1

I need P(0) --> We know that at t=0 P=100

So **P(0) = 100**

**e^(i*pi) - correct**

Step 2

I need k

I have

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

**So k=ln4.2**

**e^(i*pi) - correct (although the second equation is superfluous)**

Step 3

Write the expression by plugging in P(0) and k

P(t) = 100 * e^[(ln4.2) *t]

**e^(i*pi) - correct**

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?

Thanks