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The function known as arctanh is shorthand for the inverse hyperbolic tangent function. Hyperbolic functions are similar to trigonometric functions but are based on hyperbolas rather than circles.
They can often be useful in calculus and engineering calculations. The inverse of these functions helps us find values of the corresponding angles or areas in hyperbolic situations rather than circular ones.
The function \( anh x\) is defined as \( anh x = \frac{{e^x - e^{-x}}}{{e^x + e^{-x}}}\). Since arctanh, or \( anh^{-1}(x)\), is the inverse of \( anh(x)\), it helps to backtrack to find the original input given an output value of \( anh(x)\).
This inverse function has the range \(-1 < x < 1\) because that’s the only interval where it is defined and produces real number results. When graphed, arctanh has a shape that closely follows tanh but is flipped across the line \(y = x\).

• Arctanh allows solving for angles in hyperbolic equations.
• It is primarily used in advanced mathematics for solving integration and differential equations that are based on hyperbolic functions.

Logarithmic identities are tools that simplify the manipulation and comparison of logarithmic expressions. For instance, the exercise you encountered makes use of these identities to achieve its solution.
The identity \(\ln(a/b) = \ln(a) - \ln(b)\) allows the expression \(\frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)\) to be rewritten as \(\frac{1}{2} [\ln(1+x) - \ln(1-x)]\).
This ability to break down and separate complex logarithmic expressions is key in solving many mathematical proofs and equations.
Logarithmic properties such as power, product, and quotient rules provide a structured way to simplify the expressions themselves, often resulting in much easier arithmetic operations.

• Quotient Rule: \(\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)\)
• Product Rule: \(\ln(ab) = \ln(a) + \ln(b)\)
• Power Rule: \(\ln(a^b) = b\ln(a)\)

These identities are especially important in calculus because they simplify integration and differentiation of logarithmic functions.
They also provide a framework for proving equations like the one in the exercise, connecting inverse hyperbolic functions to logarithmic expressions.

Proof techniques are vital for verifying mathematical statements. They provide a way to demonstrate the truth of equations and inequalities logically and consistently.
When asked to prove that \(\tanh^{-1} x = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)\), a logical step-by-step approach is necessary.
The approach often begins by transforming one side of the equation to make it match the other side. In this case, rewriting one side using logarithmic identities helped simplify the expression.

• Begin by rewriting the expression using known identities or definitions.
• Be consistent with symbol manipulation; step-by-step simplification helps ensure accuracy.
• Check both sides of the equation thoroughly to ensure they are identical.

Eventually, both sides are compared to ensure they are equal in form and value. Once confirmed, the proof is complete.
Using these techniques, students can confidently solve similar proofs involving complex algebraic and calculus-based problems.