Proving the Relationship between tanh^-1(x) and ln((1+x)/(1-x))

[harrietstowe]

Homework Statement

I need to prove that tanh^-1(x) is equal to (1/2)ln((1+x)/(1-x))

Homework Equations

The Attempt at a Solution

I rewrote it as ln(x+(x²+1)^1/2)/ln(x+(x²-1)^1/2)

from here I am stuck. Thank you

 

[Dick]

Homework Statement

I need to prove that tanh^-1(x) is equal to (1/2)ln((1+x)/(1-x))

Homework Equations

The Attempt at a Solution

I rewrote it as ln(x+(x²+1)^1/2)/ln(x+(x²-1)^1/2)

from here I am stuck. Thank you

Start with y=tanh(x)=sinh(x)/cosh(x)=(e^x-e^(-x))/(e^x+e^(-x)). To find the inverse function you want to solve for x in terms of y, right? The trick is to put u=e^x, so e^(-x)=1/u. Solve for u first.

 

[harrietstowe]

Thanks so much, tried your way and worked out great