This question is probably going to have a duplicate, but my main doubt is specifically on a particular question, so please consider reading the whole thing.
First of all the function $y= \ln x$ seems (at least to me) to give infinitely many outputs on any input if complex numbers are 'allowed'. $$\ln a = \ln a + 2n \pi i (?)$$ For all integers $n$. So, even when $x$ is real, $\ln x$ is not 'unique'. This is usually answered by people over the internet in two broad ways:
$(1)$ Logarithm IS a multi-valued function, accept it and move on..
There is no 'best' value.
$(2)$ For convenience, one may define a 'principal' value or take the best value that would work in a particular situation.
But I don't understand how a universal 'best' value will be defined for an input, especially when a problem dealing with real variables is converted to complex numbers. For example: Consider $$\int_{0}^{1} \ln (1+x^3) dx = \ln 4 -3 + \frac {\pi}{\sqrt 3} $$ Right now, the logarithm has real inputs, and will give out real values, and will only give one unique value. No problem yet. Though other methods will exist (which is not my doubt), someone may consider doing: $$\int_{0}^{1} \ln (1+x) + \ln (\omega + x) + \ln (\omega ^2 + x) dx$$ And evaluate the three integrals accordingly. At some point , they will have the expressions $\ln (1+ \omega)$ and $\ln (1+ \omega ^2)$ and $\ln (\omega ^2)$ And some others
Now, is $\ln (1+ \omega^2) = \frac {5 \pi i}{3}$ or $\frac {-\pi i}{3}$. Similarly, what are all the other expressions equal to? Here, there is no convention that we can blindly follow and there doesn't seem to be one clear 'best' way to do it.
Also, since the problem originally had nothing to do with complex numbers, it should have just one answer (which it does).
But by putting different values for the above expressions, some combination will indeed give a correct answer while some combination will not.
So, what is the 'best' output for a logarithm? Since there's probably none, what kind of convention is most likely to give correct answers in such situations (where a real problem is converted to complex numbers) and why does it work?
The exponential function is injective over $\mathbb{R}$ but it is not injective over $\mathbb{C}$. Therefore, if you want to define the logarithm as its inverse, you need to select a subset of $\mathbb{C}$ where the exponential function is injective. In this sense, it is not entirely correct to say that $\log$ is a multivalued function, what you have is different logarithm functions depending on the choice of a set where $\exp(z)$ is injective.
So, regarding you question, when you are given an expression involving $\log z$, either the proper logarithm branch is already given, or you should choose one. There is never an ambiguity with respect to the values of $\log$.
[edit]
Also consider that the equality $$ \log (1+x^3) = \log(1+x)+\log(\omega +x) + \log(\omega^2+x) $$
does not hold for every branch of $\log$.