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Math Help - Logs Problem

  1. #1
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    Logs Problem

    $N_{A}(t) = 10000 + 1000t$
    $N_{C}(t) = 8000 + 3 \cdot 2^t$

    Could someone please explain how $N_{A}(t) = N_{C}(t)$ if and only if $t=\displaystyle{\frac{1}{\log_{10}2}}\cdot( 3 + \log_{10}(\displaystyle{\frac{2 + t}{3})})$?
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  2. #2
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    Re: Logs Problem

    Quote Originally Posted by Fratricide View Post
    $N_{A}(t) = 10000 + 1000t$
    $N_{C}(t) = 8000 + 3 \cdot 2^t$

    Could someone please explain how $N_{A}(t) = N_{C}(t)$ if and only if $t=\displaystyle{\frac{1}{\log_{10}2}}\cdot( 3 + \log_{10}(\displaystyle{\frac{2 + t}{3})})$?
    $10000+1000t=8000+3\cdot 2^t$

    $2000+1000t=3\cdot 2^t$

    $1000(2+t)=3\cdot 2^t$

    $3+\log_{10}(2+t)=\log_{10}(2)t+\log_{10}(3)$

    $3+\log_{10}(2+t)-\log_{10}(3)=\log_{10}(2)t$

    $3+\log_{10}\left(\dfrac{2+t}{3}\right)=\log_{10}( 2)t$

    $t=\dfrac{3+\log_{10}\left(\dfrac{2+t}{3}\right)}{ \log_{10}(2)}$
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