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Math Help - Exponential growth

  1. #1
    DBA
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    Exponential growth

    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

    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

    Step 3
    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?
    Thanks
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  2. #2
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by DBA View Post
    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
    Those two expression are not equal, as your working seems fine I would say the book is in error

    You can simplify using the log power law

    e^{ln(4.2) \cdot t} = e^{ln(4.2^t)} = 4.2^t

    \therefore \: \: P(t) = 100 \cdot 4.2^t
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