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Natural log equation

  1. Jul 31, 2008 #1
    1. The problem statement, all variables and given/known data

    sqrt(ln(x)) = ln(sqrt(x))

    2. Relevant equations

    3. The attempt at a solution

    I've been trying to do this for some time now. Could anyone give some tips on how to get started with this?
  2. jcsd
  3. Jul 31, 2008 #2
    Try squaring both sides and then bring the ln over.
  4. Jul 31, 2008 #3
    (ln(sqrt(x))^2 - ln x = 0

    2 * ln(sqrt(x)) - ln x = 0

    ln (sqrt(x)) (2-ln(sqrt(x))) = 0

    x=1 and e^4.

    Ok got it now. I tried this for 10+ minutes and was totally lost. Somehow brain starts to work a little after posting here :P
    Last edited: Jul 31, 2008
  5. Aug 1, 2008 #4
    Haha yeah happens sometimes.

    Btw, what do u mean by:
    2 * ln(sqrt(x)) - ln x = 0

    this is what i did:

    After squaring:
    ln(x) = ln (x^1/2)*ln(x^1/2)
    Bring the half down:
    ln(x) = 1/2(ln(x))*1/2(ln(x))
    then after a bit of rearranging:
    1/4(ln(x)ln(x)) - ln(x) = 0.
    Factor the ln(x) out.
    ln(x)[1/4(lnx) -1] = 0.
    ln x = 0 or 1/4lnx = 1
    x = e^0
    =1 or lnx = 4, x = e^4
  6. Aug 1, 2008 #5
    After the first line above, the rest of your manipulations don't seem valid. [ln(sqrt(x))]^2 does NOT equal 2 * ln(sqrt(x)). 2 * ln(sqrt(x)) = ln[(sqrt(x))^2]. See the difference?

    Similarly ln(sqrt(x)) * ln(sqrt(x)) does NOT equal ln(x). Only the addition of two logs with the same base allows for the arguments to be multiplied together. I think you need to check your rules again.
  7. Aug 1, 2008 #6


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    [tex]\sqrt{ln(x)}= ln(\sqrt{x})[/tex]

    I think I would have been inclined to write this as
    [tex]\sqrt{ln(x)}= (1/2)ln(x)[/tex]
    and let u= ln(x) so I have
    [tex]\sqrt{u}= (1/2)u[/tex]
    or u= u2/4. Then, then, is equivalent to u2- 4u= 0 which has u= 0 and u= 4 as solutions. Since u= ln(x), u= 0 gives x= 1 and u= 4 gives u= e4.
  8. Aug 1, 2008 #7
    Yea i made a mistake there. I was fixing it yesterday but it said can't edit after 1 hour. This is how I did it.

    square both sides and factor ln(sqrt(x)) out.

    ln x = (ln sqrt(x))^2

    ln(sqrt(x)) (ln(sqrt(x)) - 2) = 0

    ln(sqrt(x)) = 0 and ln(sqrt(x)) - 2 = 0

    x=1 x=e^4

    It's good to see different ways of doing it :)
    Thanks for the help, please tell me if there's any mistake in this.
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