The formula is true for some constraints on the values of a and b.
First, a>0.
Second, a+b>0 or a>-b.
Third, 1+b/a>0 follows from the precedent two constraints.
Considering the truth of these inequalities, it may be elementary proved (knowing the properties of the function Log[]) that:
Log[a]+Log[1+b/a]=Log[a]+Log[(a+b)/a]=Log[a*(a+b)/a]=Log[a+b]
In the Wolfram Mathematica the truth of the above formula may be verified by applying the function Reduce[]:
In[1]:= Reduce[Log[a + b] == Log[a] + Log[1 + b/a], {a, b}, Reals]
Out[1]= a > 0 && b > -a
