I know that the logarithm function can be defined in a field, as an inverse of the exponential function (from here), if the exponential function is defined such that it is bijective, but I'd like to know if the common logarithm laws e.g. $$log_abc=log_ab+log_abc$$ $$log_ab^c=c*log_ab$$ also apply, where $a,b,c\in F$, where $F$ is a field. It seems to me that they can apply: for example, let $log_ab^c=x$ where $a^x=b^c$. Then, we have (basically proof for this law in the reals):$$log_ab=c$$ $$\therefore a^c=b$$ $$\therefore (a^c)^n=b^n$$ $$\therefore a^{cn}=b^n$$ $$\therefore log_ab^n=cn=n*log_ab$$ It seems to me that the above manipulations don't violate any of the field axioms, or indeed ring axioms, provided that n is less than the number of elements. Thus, I'm wondering if it may be possible for all the log laws to be satisfied in any division ring.
I'm currently a high school student, so I would appreciate relatively simple answers if possible, and also pointers on where my logic may be wrong. If not possible, I'll do my best to understand.