To Compute log(a+b)

I want the complete expansion of log(a+b)=?

for ex

log(a*b)=log a + log b;

log(a/b)=lob a – log b;
Similar to this, is there any expansion for log(a+b)???

What does “log*” mean?

I have come across the term O(log* N) in a book I’m reading on data structures. What does log* mean? I cannot find it on Google, and WolframAlpha doesn’t understand it either.