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Thread: Logarithm help!

  1. #1
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    Logarithm help!

    Need help with problem.
    Log (x-1)-log (x+1)=3-log (x-2)

    I know the answer is about 1003.994 but need help figuring out how to get the answer. I got to (-1)(x-2)=1000x+1000 but don't know where to go from there.
    Last edited by Nfalconer; Jul 29th 2018 at 12:31 PM.
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  2. #2
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    Re: Logarithm help!

    Quote Originally Posted by Nfalconer View Post
    Log (x-1)-log (x+1)=3-log (x-2)
    I know the answer is about 1003.994
    Says who?
    Answer is ~23.844

    Havva look:
    Wolfram|Alpha: Computational Intelligence)
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  3. #3
    MHF Contributor MarkFL's Avatar
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    Re: Logarithm help!

    We are given to solve:

    $\displaystyle \log(x-1)-\log(x+1)=3-\log(x-2)$

    Rewrite using log rules (and assuming logs are base 10)

    $\displaystyle \log\left(\frac{x-1}{x+1}\right)=\log\left(\frac{1000}{x-2}\right)$

    This implies:

    $\displaystyle \frac{x-1}{x+1}=\frac{1000}{x-2}$

    $\displaystyle (x-1)(x-2)=1000(x+1)$

    $\displaystyle x^2-3x+2=1000x+1000$

    $\displaystyle x^2-1003x-998=0$

    Apply the quadratic formula...what do you get?
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: Logarithm help!

    Quote Originally Posted by DenisB View Post
    Says who?
    Answer is ~23.844

    Havva look:
    Wolfram|Alpha: Computational Intelligence)
    W|A assumes a base of e when none is given.
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  5. #5
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    Re: Logarithm help!

    Quote Originally Posted by Nfalconer View Post
    Need help with problem.
    Log (x-1)-log (x+1)=3-log (x-2)

    I know the answer is about 1003.994 but need help figuring out how to get the answer. I got to (-1)(x-2)=1000x+1000 but don't know where to go from there.
    You are using base 10, see here.

    @Nfalconer, I have noticed that you use older norms in mathematics. In today's practice $\log(x)$ is taken to be base $e$ or the natural. In fact $\log_b(x)=\dfrac{\log(x)}{\log(b)}$.

    So $ \log_{10} (x-1)-\log_{10} (x+1)=3-\log_{10} (x-2)$ translates to $\log_{10}\left(\dfrac{(x-1)(x-2)}{(x-1)}\right)=3$
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    Re: Logarithm help!

    Quote Originally Posted by MarkFL View Post
    W|A assumes a base of e when none is given.
    Which is standard practice in mathematics today.
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  7. #7
    MHF Contributor MarkFL's Avatar
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    Re: Logarithm help!

    Quote Originally Posted by Plato View Post
    Which is standard practice in mathematics today.
    Yes, when I was a student, at the elementary level, a log without a base given was assumed to be base 10 and ln() (natural log) was used for base e logs. Then when you got into 3rd year course at the university level, such as course in analysis, log() was used for base e logs.
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    Re: Logarithm help!

    After using the quadratic formula I did not get 23.844. Please show me how you got that.
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    Re: Logarithm help!

    Quote Originally Posted by Nfalconer View Post
    After using the quadratic formula I did not get 23.844. Please show me how you got that.
    That's because he used log base e instead of log base 10 as we pretty much assumed by the problem as written. 23.844 is not correct if we use log base 10.

    -Dan

    Addendum: Personally I still use ln for log base e. I really don't know why this has been "abandoned."
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  10. #10
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    Re: Logarithm help!

    Quote Originally Posted by Nfalconer View Post
    After using the quadratic formula I did not get 23.844. Please show me how you got that.
    Read carefully reply #3. then look at THIS.
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  11. #11
    MHF Contributor MarkFL's Avatar
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    Re: Logarithm help!

    In some of the computer science courses I took a log without a stated base was assumed to be base 2. That made sense though, obviously.
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  12. #12
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    Re: Logarithm help!

    Quote Originally Posted by Nfalconer View Post
    Need help with problem.
    Log (x-1)-log (x+1)=3-log (x-2)

    I know the answer is about 1003.994 but need help figuring out how to get the answer. I got to (-1)(x-2)=1000x+1000 but don't know where to go from there.
    Here is the Answer based on base e.

    $\displaystyle \log\left(\frac{x - 1}{x+1}\right) + \log (x -2)= 3$
    $\displaystyle \log\left(\frac{(x - 1)(x-2)}{x+1}\right) = 3$
    $\displaystyle \left(\frac{(x-2)(x-1)}{x+1}\right) = \log^{-1}(3)$
    $\displaystyle \frac{(x-2)(x-1)}{x+1}=20.08554$
    $\displaystyle x^2 - 3x + 2 =20.08554x + 20.08554$
    $\displaystyle x^2 - 23.08554x - 18.08554 = 0$

    $\displaystyle x =314.1849285458 $
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    Re: Logarithm help!

    Quote Originally Posted by Nfalconer View Post
    Need help with problem. Log (x-1)-log (x+1)=3-log (x-2)
    I know the answer is about 1003.994 but need help figuring out how to get the answer.
    Quote Originally Posted by x3bnm View Post
    Here is the Answer based on base e.
    $\displaystyle x =314.1849285458 $
    @x3bnm, isn't it clear to you that the OP clearly implies that it is about $log_{10}~?$
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  14. #14
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    Re: Logarithm help!

    Quote Originally Posted by Plato View Post
    @x3bnm, isn't it clear to you that the OP clearly implies that it is about $log_{10}~?$
    Sorry for the misunderstanding. But how did he come up with 1003.994? So I was confused. Also, he didn't specifically mention about the base. So I thought why not show him another way. Sorry Plato I didn't mean to offend anybody.
    Last edited by x3bnm; Jul 29th 2018 at 08:07 PM.
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  15. #15
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    Re: Logarithm help!

    Quote Originally Posted by Nfalconer View Post
    Need help with problem.
    Log (x-1)-log (x+1)=3-log (x-2)

    I know the answer is about 1003.994 but need help figuring out how to get the answer. I got to (-1)(x-2)=1000x+1000 but don't know where to go from there.

    $\displaystyle \log(x-1)-\log(x+1)=3-\log(x-2)$

    $\displaystyle \log\left(\frac{x - 1}{x+1}\right) + \log (x -2)= 3$

    $\displaystyle \log\left(\frac{(x - 1)(x-2)}{x+1}\right) = 3$

    $\displaystyle \left(\frac{(x-2)(x-1)}{x+1}\right) = \log^{-1}(3)$

    $\displaystyle \frac{(x-2)(x-1)}{x+1}=1000$

    $\displaystyle x^2 - 3x + 2 =1000x + 1000$

    $\displaystyle x^2 -1003x -998 = 0$

    $\displaystyle x = \frac{1003 \pm \sqrt{1003^{2} - 4(-998)}}{2}$


    $\displaystyle x = 1002.003995988$ [Answer]

    You can check the quadratic equation below:

    Wolfram|Alpha: Computational Intelligence

    if you use a calculator the answer will be 1003.994 So you are right.
    Last edited by x3bnm; Jul 29th 2018 at 08:47 PM.
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