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Math Help - Log System of equations

  1. #1
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    Log System of equations

    I'm needing a little assistance with this problem, haven't any trouble until this.

    A village of 1000 inhabitants increases at a rate of 10% per year. A neighbouring village of 2000 inhabitants decreases at a rate of 5% per year. After how many years will these two villages have the same population?

    So I wrote them as two exponential functions and made them equal to each other. Not sure this is how I should do it but it feels right.
    1000(1.10)^x=2000(0.95)^x
    Then by definition C^x=y <==> log(y)=x

    Turned them into this
    log(1.1)1000=log(0.95)2000
    then I know it can be turned into this.
    log1000/log1.1=log2000/log0.95

    And now I'm stuck, help!
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  2. #2
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    Quote Originally Posted by Gerard View Post
    So I wrote them as two exponential functions and made them equal to each other. Not sure this is how I should do it but it feels right.
    1000(1.10)^x=2000(0.95)^x
    To make things easier, divide both sides by 1000.

    What next?
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  3. #3
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    Quote Originally Posted by pickslides View Post
    To make things easier, divide both sides by 1000.

    What next?
    (1.1)^x=2(0.95)^x

    Then using the definition...
    log1/log1.1 = log2/log0.95


    ?
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  4. #4
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    Using what definition? Taking the logarithm of both sides (any base) of
    (1.1)^x= 2(0.95)^x you get

    x log(1.1)= x log(.95)+ log 2
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