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Homework Help: Show that x=tanh(y) is

  1. Apr 10, 2014 #1
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution
    I know that x=tanh(y) can be shown as y=tanh^-1(x). The problem is how do i get from there to the next part. I'm kinda stuck here.

    I can show that
    x = sinh(y)/cosh(y)
    x = (e^y - e^-y) / (e^y + e^-y)

    x = lny (1+y)/(1-y)
    I know something is terribly wrong in the last step. and I dont think this is the way to proceed with the question. Can you help me out?

    Attached Files:

  2. jcsd
  3. Apr 10, 2014 #2


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    Homework Helper

    Up to here, it's correct. I assumed you cancelled the 1/2 from top and bottom.

    No idea how you made the jump to the last step. No law of logs allows that.

    Hint: let z = ey and solve algebraically for z in terms of x. Then take the log of both sides to get y in terms of x.

    Remember that e-y is the reciprocal of ey.
  4. Apr 10, 2014 #3
    x = (ey - e-y)/(ey + e-y)
    Another easy way to finish from there is:
    xey + xe-y = ey - e-y
    and then multiply each of the 4 terms by ey,
    which will lead (after a couple of steps) to
    (x-1) e2y = (-1-x)
    and the rest is quite easy.
  5. Apr 11, 2014 #4
    So far I have got x = (z^2 - 1) / (z^2 +1)
    Im stuck from there though.

    I was able to get the answer using az_lender method but I want to know how to do it using yours.
    Can you guide me from where I am stuck? thank you
  6. Apr 11, 2014 #5


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    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    It's pretty much the same sequence of steps.
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