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Math Help - Applications of Exponential Equation Question

  1. #1
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    Applications of Exponential Equation Question

    Hi everyone,

    Recently I have been working with applying exponential equations to the real world. However, one question completely threw me as is laid out below:

    Q. The population of a town, N, is modeled by the function N = 15000(2 ^ 0.01t), where "t" is the number of years since 1980. In what year will the population reach 20000 people?

    A. Okay, well to start off with, the general form of an exponential equation is:

    y = ab ^ x

    I know that "a" is the initial value and in this case is 15000.

    N = 15000(2 ^ 0.01t)


    I substitute N for 20000

    20000 = 15000(2 ^ 0.01t)

    Then do I divide both sides by 15000? And Once I have done that, and have transformed the equation into logarithmic form, how do I solve for "t"? Normally, it's quite easy, but in this case, the power of "t" is prefixed with "0.01", and I am unsure how to work with that.

    Any help on solving this will be much appreciated!

    Thanks,

    Nathaniel.
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  2. #2
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    The given problem is in the form

    y = ab^{xt}

    Take log on both side

    ln(y/a) = xt*ln(b)

    Substitute the values and find t.
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  3. #3
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    Quote Originally Posted by BinaryBoy View Post
    Hi everyone,

    Recently I have been working with applying exponential equations to the real world. However, one question completely threw me as is laid out below:

    Q. The population of a town, N, is modeled by the function N = 15000(2 ^ 0.01t), where "t" is the number of years since 1980. In what year will the population reach 20000 people?

    A. Okay, well to start off with, the general form of an exponential equation is:

    y = ab ^ x

    I know that "a" is the initial value and in this case is 15000.

    N = 15000(2 ^ 0.01t)


    I substitute N for 20000

    20000 = 15000(2 ^ 0.01t)

    Then do I divide both sides by 15000? And Once I have done that, and have transformed the equation into logarithmic form, how do I solve for "t"? Normally, it's quite easy, but in this case, the power of "t" is prefixed with "0.01", and I am unsure how to work with that.
    Once you have solved for 0.01t, divide both sides by 0.01!

    Any help on solving this will be much appreciated!

    Thanks,

    Nathaniel.
    Follow Math Help Forum on Facebook and Google+

  4. #4
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    Thanks for the help so far, sa-ri-ga-ma and HallsofIvy. Much appreciated.

    The answer should be 2022, but I am not getting that. Am I supposed to divide the log of 20000 by the log of 15000, or just log(20000 / 15000)?

    And from there, do I then divide the result by the log of 2?
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  5. #5
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    Your approach is correct.
    ln(20/15) = 0.01*t*ln(2)

    Solve for t. You will get 41.5 ~ 42. Add this to 1980 to get the final result.
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