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Inverse hyperbolic tangent

The inverse hyperbolic tangent is written with logarithms as

Area functions can be described with logarithms because all hyperbolic functions are defined by exponential functions.

 

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Explanation

We start with the hyperbolic tangent to which applies

Substitution of e^θ = k gives

We continue only with the positive solution and substitute k = e^θ so that

Now we take the logarithm on both sides

Substitution of θ = artanh x gives

 

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Example 1

You can see that tanh^-1(0) = 0, because

 

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