SUMMARY
The discussion clarifies the mathematical identity involving the hyperbolic tangent function and natural logarithm. The correct expression is \(\frac{x-1}{x+1} = \tanh\left(\frac{\ln x}{2}\right)\), not \(\frac{x-1}{x+1} = \tanh\left(\ln\left(\frac{x}{2}\right)\right)\). This correction stems from the identity \(\text{artanh}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)\), which can be derived by substituting \(y = \frac{1+x}{1-x}\) and solving for \(w\) in the equation \(z = \tanh(w)\).
PREREQUISITES
- Understanding of hyperbolic functions, specifically \(\tanh\) and \(\text{artanh}\)
- Familiarity with natural logarithms and their properties
- Basic algebraic manipulation skills
- Knowledge of mathematical identities and their derivations
NEXT STEPS
- Study the derivation of the identity \(\text{artanh}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)\)
- Explore applications of hyperbolic functions in calculus and physics
- Learn about the properties and applications of logarithmic functions
- Investigate further mathematical identities involving hyperbolic and logarithmic functions
USEFUL FOR
Mathematicians, students studying calculus or advanced algebra, and anyone interested in the properties of hyperbolic functions and logarithms.