FullSimplify not yielding True for Log[a/b] == Log[a] - Log[b]?

Question

I am not receiving True for Log[a/b]==Log[a]-Log[b]. Is there a way around this problem?

Is this statement true always? What happens when b is complex, or b == 0? — DrMrstheMonarch, May 29 '22 at 23:25
@DrMrstheMonarch Oh good point! Is there a way to add conditions to FullSimplify? — MathIsHard, May 29 '22 at 23:26
It is sufficient to use Simplify, e.g. Simplify[Log[a/b] == Log[a] - Log[b], a != 0 && b > 0] — Artes, May 29 '22 at 23:32
@Alan: This is a weakness/bug in PowerExpand and the documentation says it in a similar situation "Expand a power of a product; the result may not be correct everywhere:" — user64494, May 30 '22 at 05:23
@user64494 It is a feature not a weakness or bug. But yes, care in use is required. In this case, it meets the demands of the OP. — Alan, May 30 '22 at 16:26

Alexei Boulbitch · Accepted Answer · 2022-05-30T13:10:54.410

One can use a standard Mma approach of working with logarithms. It is PowerExpand.

PowerExpand treats its expressions assuming all variables are positive. Normally

Log[ab] == Log[a] + Log[b]

is not always true, e.g.

0 = Log[(-1) (-1)] != Log[-1] + Log[-1] == 2 Pi*I

One can use Simplify together with the declaration of the variables as positive (if it is true):
```
 Simplify[Log[a] + Log[b], Assumptions -> a > 0 && b > 0]
(* Log[a b]  *)
```
PowerExpand[Log[a*b], Assumptions -> a > 0 < b]
```
 (* Log[a] + Log[b]  *)
```

In general

PowerExpand[Log[a b], Assumptions -> Element[{a, b}, Complexes]]
(*  2 I [Pi] Floor[1/2 - Arg[a]/(2 [Pi]) - Arg[b]/(2 [Pi])] + Log[a] + Log[b]  *)

If I know that the variables are positive, and want to transform logarithms I use the following functions:

expandLog[expr_] := Module[{rule1, rule2, a, b, x, g}, rule1 = Log[a_b_] -> Log[a] + Log[b]; rule2 = Log[a_^x_] -> xLog[a]; g[x_] := (x /. rule1) /. rule2; FixedPoint[g, expr] ];

and

collectLog[expr_] := Module[{rule1a, rule1b, rule2, g, a, b, x},
       rule1a = Log[a_] + Log[b_] -> Log[a*b];
       rule1b = Log[a_] - Log[b_] -> Log[a/b];
       rule2 = x_*Log[a_] -> Log[a^x];
       g[x_] := x /. rule1a /. rule1b /. rule2;
       FixedPoint[g, expr]
       ];

They enable me to transform logarithms according to my wish. Try them.

Have fun!

FullSimplify not yielding True for Log[a/b] == Log[a] - Log[b]?

1 Answers1