how do you factor a out of ln(a+b)?

Question

I am trying to get
ln(a+b)=ln(a)+(b/a)
by approximating that ln(1+x)=x when x<<1

Answer 1

ln(a + b) is the same as

ln(a(1 + b/a))

Which is the same as

ln(a) + ln(1 + b/a)

That's the best you can do when factoring out an a, and it isn't even factoring out an a.

Answer 2

Well, first of all, you can't factor that. Second, even though, very close to 0, ln(1+x) is very close to x, what does that have to do with ln(a+b)? If a and b can be anything, there's no reason to think we're anywhere close to 0.

Let a = 2 and b = 3. ln(a+b) = ln 5 = 1.609, but ln(a) + b/a = ln 2 + 3/2 = 0.6931+1.5 = 2.1931, which is not particularly close.