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Math Help - solving quadratic equation involving log

  1. #1
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    solving quadratic equation involving log

    Hi,
    How do you solve an equation with a quadratic expression on one side and a log expression on the other?
    The specific question is:

    2*n*log10(n) = 0.1*n^2

    (Background: this is actually for an algorithm analysis question - I am trying to find the intersection points of the worst-case running times of two algorithms)

    Thanks
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  2. #2
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    Re: solving quadratic equation involving log

    Well if you write this as

    \displaystyle \begin{align*} 2n\log_{10}{n} &= 0.1n^2 \\ 0 &= 0.1n^2 - 2n\log_{10}n \\ 0 &= n(0.1n - 2\log_{10}{n}) \end{align*}

    Since \displaystyle \log_{10}{0} is undefined, we can't accept \displaystyle n = 0 as a solution, which means we have

    \displaystyle 0 = 0.1n - 2\log_{10}{n}

    You will need to use a numerical method to solve this.
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  3. #3
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    Re: solving quadratic equation involving log

    This is a tricky one, but maybe we can work it around a bit so the solution might be easier to find.

    \displaystyle 2n\log_{10}n = 0.1n^2

    \displaystyle \log_{10}n = \frac{0.1n^2}{2n}

    \displaystyle \log_{10}n = 0.05n

    \displaystyle n = 10^{0.05n}

    You can solve it numerically from here.
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  4. #4
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    Re: solving quadratic equation involving log

    Thanks a lot for the fast replies.
    I also got up to n = 10^(0.05*n) and did not know how to proceed any further. What is meant by solving it numerically, exactly? Does it basically mean a brute-force approach?
    Cheers
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  5. #5
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    Re: solving quadratic equation involving log

    Yep, brute force, open up a spreadsheet you'll find n \approx 29
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    Re: solving quadratic equation involving log

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