tommy's uniqueness conditions

[bo198214]

now we introduce "another slog"

" another slog " = slog(x + v(x))

v(x) is meromorphic on C and has period 1 and v(0) = 0.

( we will later see that v(x) must be meromorphic rather than just entire )

so

slog(exp(x + v(x))) = slog(x + v(x)) + 1

letting y = x + v(x) we see that this is also a solution.

Dont know exactly how you arrive at that equation.
say slog2(x)=slog(x+v(x))
then slog2(exp(x))=slog(exp(x) + v(exp(x)))
I dont see why slog2 should be another slog.

sexp2(x)=sexp(x+v2(x)) that works, because:
sexp2(x+1)=sexp(x+1+v2(x+1))=sexp(x+v(x)+1)=exp(sexp(x+v(x))=exp(sexp2(x)).

For the slog one would choose
slog3(x)=slog(x)+v(slog(x))
to get another slog.

It seems you based your conclusions on a wrong assumption.