Answer
$$\coth^{-1}\Big(\frac{5}{4}\Big)=\ln3$$
Work Step by Step
We have $$\coth^{-1}x=\frac{1}{2}\ln\frac{x+1}{x-1}$$ for $|x|\gt1$
Therefore, $$\coth^{-1}\Big(\frac{5}{4}\Big)=\frac{1}{2}\ln\frac{5/4+1}{5/4-1}$$ $$=\frac{1}{2}\ln\frac{9/4}{1/4}=\frac{1}{2}\ln9$$ $$=\ln9^{1/2}=\ln3$$
