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Solve for x logarithm question

  1. Jan 13, 2013 #1
    1. The problem statement, all variables and given/known data
    The problem asks to solve for x

    log(x)=-4x+5 (the log has base 10 - not natural log)


    2. Relevant equations



    3. The attempt at a solution
    log(x)=-4x+5

    10^-4x+5=x

    log(10^-4x+5)=log(x)
    -4x+5*log(10)=log(x)
    -4x+5=log(x)

    seems at first to be a simple enough question to solve, however I only end up with a circular solution bringing me back to the original question when i take the log of this to bring down the exponent (of course)


    Any help would be appreciated :)
     
  2. jcsd
  3. Jan 13, 2013 #2

    CAF123

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    You could try writing $$log_{10}x=\frac{ln x}{ln10} $$
     
  4. Jan 13, 2013 #3

    HallsofIvy

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    There isn't going to be any simple "algebraic" way to solve that because you have the unknown, x, both inside and outside the logarithm. The first thing I would do is use CAF123's suggestion to change to natural logarithm: [itex]ln(x)/ln(10)= -4x+ 5[/itex] so that [itex]ln(x)= (-4/ln(10))x+ 5/ln(10)[/itex]. Now take the exponential of both sides:
    [itex]x= e^{(-4ln(10))x}e^{5/ln(10)}[/itex]

    [itex]xe^{(4/ln(10))x}= e^{5/ln(10)}[/itex]

    Now let y= (4/ln(10))x so that x= (ln(10)/4)y and the equation becomes
    [itex](ln(10)/4)ye^y= 5^{5/ln(10)}[/itex]

    [tex]ye^y= \frac{5^{5/ln(10)}}{ln(10)/4}[/tex]

    and therefore
    [tex]y= W\left(\frac{5^{5/ln(10)}}{ln(10)/4)}\right)[/tex]
    where "W" is Lambert's W function (http://en.wikipedia.org/wiki/Lambert_W_function) which is defined as the inverse function to [itex]f(x)= xe^x[/itex]
    (I told you there was no simple "algebraic" solution!)

    Since we defined y= (4/ln(10))x, we have, finally,
    [tex]x= ln(10)\frac{W\left(\frac{5^{5/ln(10)}}{ln(10)/4)}\right)}{4}[/tex]
    May you have joy of it!

    Since this was posted under "Precalculus Mathematics", where in the world did you get this problem?
     
  5. Jan 13, 2013 #4

    CAF123

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    Should that be ##ln(x) = ln(10)[5 - 4x]?## Instead of taking the exponential of both sides, I then said ##ln(x) = 5ln(10) - 4x ln(10)\,\Rightarrow ln(x) = ln \left(\frac{10^5}{10^{4x}} \right)##. I now take exp to get ##x10^{4x} = 10^5.##
     
  6. Jan 13, 2013 #5
    it will be easier to solve this question using Newton-Raphson method. :)
     
  7. Jan 13, 2013 #6

    HallsofIvy

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    Yes, I messed that up. Thanks for catching it. But that still has no "algebraic" solution. In order to use the W function, you will need to convert to "e". [tex]10^{4x}= e^{ln(10^{4x}}= e^{4xln(10)}[/tex] so the equation is the same as [tex]xe^{4xln(10)}= 10^5[/tex].

    Let y= 4x ln(10) so that x= y/(4 ln(10)). The equation becomes, in terms of y, [tex]\frac{ye^y}{4ln(10}= 10^5[/tex]. [tex]ye^y= 400000 ln(10)[/tex], [tex]y= W\left(40000 ln(10)\right)[/tex].

    Then [tex]x= \frac{W\left(40000 ln(10)\right)}{4 ln(10)}[/tex]
     
  8. Jan 13, 2013 #7

    ehild

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    You can solve the equation numerically by iteration.
    Rewrite it as

    x=(5-log(x))/4,

    Get the next approximation from a previous one by xk+1=(5-log(xk))/4.

    Using x1=1, substitute and get the next approximation: x2=1.25. Substitute again, now you get x3=1.227 .....
     
  9. Jan 16, 2013 #8

    HallsofIvy

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    Well, sure, if you want to do it the easy way!
     
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