[?] Simplify the logarithm of a sum: Log[a+b]?

[silvaalberto (Alberto Silva Ariano)]

Dear Wolfram team and community:

How can I rearrange the logarithm of a sum: Log[a+b]

Is the following formula (found on wikipedia) true or false? :

Log[a+b] = Log[a] + Log[1 + b/a]

If is true, how can I do it in Mathematica?

[vaungureanu (Valeriu Ungureanu)]

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

[sghuisman (Sander Huisman)]

Reduce might ‘reduce’ it a bit too much:

RegionPlot[
 Chop[Log[a] + Log[1 + b/a] - Log[a + b]] == 0, {a, -20, 20}, {b, -20,
   20}]

While your inequality is:

For complex number the inequality becomes much more complex…