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

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SUMMARY

The relationship between tanh-1(x) and (1/2)ln((1+x)/(1-x)) is established through the inverse hyperbolic tangent function. The proof begins with the definition y = tanh(x) = sinh(x)/cosh(x) = (ex - e-x)/(ex + e-x). By substituting u = ex and manipulating the equation, one can derive the desired logarithmic expression. This method effectively demonstrates the equivalence of the two expressions.

PREREQUISITES
  • Understanding of hyperbolic functions, specifically tanh, sinh, and cosh.
  • Familiarity with logarithmic properties and transformations.
  • Basic knowledge of exponential functions and their inverses.
  • Ability to manipulate algebraic expressions involving variables.
NEXT STEPS
  • Study the derivation of inverse hyperbolic functions, focusing on tanh-1(x).
  • Learn about the properties of logarithms, particularly in relation to hyperbolic functions.
  • Explore the relationship between exponential functions and their inverses in greater depth.
  • Practice solving equations involving hyperbolic and logarithmic identities.
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Students studying calculus, particularly those focusing on hyperbolic functions and their applications, as well as educators seeking to explain the relationship between inverse hyperbolic functions and logarithmic expressions.

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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
 
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harrietstowe said:

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.
 
Thanks so much, tried your way and worked out great
 

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