Complex conjugate of the logarithm of the hyperbolic tangent

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Given the Schwarz reflection principle, I would aytomaticaly write down that the complex conjugate of the following function: $$ ln[tanh(z)] $$, where z is a complex number, is: $$ln[tanh(\bar{z})] $$. Nevertheless, the result I got from Mathematica, is totally different: $$ln[-itan(z)] $$. How can this be correct? Can conjugation change the function?