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Math Help - Natural logarithms

  1. #1
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    Natural logarithms

    Hi,

    I am doing a review on logarithms. When I reached my last question's solution I found something strange.

    Question: Solve, Ln(y-1)=1+Ln(3y+2)

    The solution gives:
    Step 1. Ln(y-1)-Ln(3y+2)=1
    Ln((y-1)/(3y+2))=1

    Step 2. e^( Ln((y-1)/(3y+2))=e^1

    (y-1)/(3y+2)=e

    The solution then goes on:

    y-1=e(3y+2)=3e*y+2e

    (1-3e)y=1+2e

    Finally,

    y=(1+2e)/(1-3e)=-0.8996


    Can you tell me what's wrong here and what does e mean without an index?


    Or is there something I have failed to recognise?
    Last edited by charlesrussell; October 4th 2012 at 11:06 AM. Reason: Resolved issue
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  2. #2
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    Re: Natural logarithms

    Quote Originally Posted by charlesrussell View Post
    Hi,

    I am doing a review on logarithms. When I reached my last question's solution I found something strange.

    Question: Solve, Ln(y-1)=1+Ln(3y+2)

    The solution gives:
    Step 1. Ln(y-1)-Ln(3y+2)=1
    Ln((y-1)/(3y+2))=1

    Step 2. e^( Ln((y-1)/(3y+2))=e^1

    (y-1)/(3y+2)=e

    The solution then goes on:

    y-1=e(3y+2)=3e*y+2e

    (1-3e)y=1+2e

    Finally,

    y=(1+2e)/(1+3e)=-0.8996

    I found no solution and after graphing saw that there could be no solution for real numbers.

    Can you tell me what's wrong here and what does e mean without an index?
    How does e become -0.404273?

    Or is there something I have failed to recognise?
    e is Euler's number, which has the property of being the base of an exponential function so that its derivative is equal to the original function.
    Thanks from charlesrussell
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  3. #3
    MHF Contributor ebaines's Avatar
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    Re: Natural logarithms

    You actually have the correct answer, with y = -0.8996 (one minor nit to pick - in the line right after "Finally" you should have y=(1+2e)/(1-3e)=-0.8996).

    The issue is that the log of a negative number is a complex number, which has a real part consisting of the log of the absoulte value of that negative number plus an imaginary part equal to pi. This comes from: log(-a) = log(-1 x a) = log (-1) + log (a), and form e^(i pi) =-1 you get log(-1) = i pi. So your solution of -0.8996 when put back into the original equation yields:

    ln(y-1) = 1 + ln(3y+2)
    ln(-1.8996) = 1 + ln(-0.69883)
    ln(1.8996)+i pi = 1 + ln(0.69883) + i pi
    0.6416 + i pi = 1 + -0.3584 + i pi
    0.6416 + i pi = 0.6416 + i pi

    So it checks out.
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  4. #4
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    Re: Natural logarithms

    what does e mean without an index?
    Since you are doing a problem involving "ln(x)", I presume you know what that means. It is the inverse function to " e^x". And that means just the number "e" to the x power. Saying that is "without an index" means it is e^1, just as "x", in a polynomial "without a coefficient" would be "1x".
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  5. #5
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    Re: Natural logarithms

    Your reply to my query is the answer I was looking for.
    Last edited by charlesrussell; October 4th 2012 at 11:13 AM. Reason: As required
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  6. #6
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    Re: Natural logarithms

    Thanks for your breakdown on complex numbers. It now makes sense to me.
    I forgot to change my settings to polar which didn't help any.
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