$\DeclareMathOperator{\arcsinh}{arcsinh}$I have seen that $$ \arcsinh(x) = \ln(x + \sqrt{x^2 + 1}) \tag{1} $$ and also that $$ \arcsinh(x/a) = \ln(x + \sqrt{x^2 + a^2}). \tag{2} $$
I have to calculate $\arcsinh(\frac{1}{2}\sqrt{2})$; the above formula give values of $\frac{1}{2} \ln 2$ and $\ln 4$, respectively.
My question is what have i done wrong/assumed. I know for certain that both formula are correct as they are in official formula booklet.
The second formula is incorrect. You are missing a term "$-\ln a$" which should be appended to the end.
If you have truly quoted it verbatim from a booklet, then the booklet is wrong.
Is it possible that the extra term was printed on a second line and you simply missed it?
As @Andrew D. Hwang hints, if these are antiderivatives, they should both have a "$+C$" tacked on, and that could absorb the missing constant; but as written, he formula is wrong.