# Show that x=tanh(y) is

1. Apr 10, 2014

### uzman1243

1. The problem statement, all variables and given/known data

2. Relevant equations
(above)

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?

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2. Apr 10, 2014

### Curious3141

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.

3. Apr 10, 2014

### az_lender

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.

4. Apr 11, 2014

### uzman1243

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

5. Apr 11, 2014

### SammyS

Staff Emeritus
It's pretty much the same sequence of steps.