How to prove that $\log a = \log b \Rightarrow a=b$ without assuming that its true?

Question

Why is $\log a = \log b \Rightarrow a=b$ true?

This "answer" does not answer the question and should be a comment. — bbgodfrey, Jun 19 '21 at 17:56
@will Is this a question abou math or about the Mathematica software? — A.G., Jun 20 '21 at 04:04

user64494 · Answer 1 · 2021-06-19T17:45:02.473

2

Straightforward approaches are

FindInstance[Exists[{a, b}, Log[a] == Log[b] && a != b], {a, b}, Complexes]
*({})*

and

FunctionInjective[Log[z], z, Complexes]
(*True)*

edited Jun 19 '21 at 17:45

answered Jun 19 '21 at 17:35

user64494

score 2 · Answer 2 · answered Jun 19 '21 at 18:04

2

Reduce can do it:

Reduce[Log[a] == Log[b], {a, b}]
(* b == a *)

answered Jun 19 '21 at 18:04

Coolwater

Or Reduce[Log[a] == Log[b], b] // FullSimplify. Simply Reduce[Log[a] == Log[b]] does not work. – user64494 Jun 19 '21 at 18:08

mikado · Answer 3 · 2021-06-19T18:35:21.613

0

Simple proof

Exp /@ (Log[a] == Log[b])
(* a == b *)

Of course the converse is not true

Log /@ (Exp[a] == Exp[b])
(* Log[E^a] == Log[E^b] *)

edited Jun 19 '21 at 18:35

answered Jun 19 '21 at 17:17

mikado

This demonstration only holds because $x\to e^x$ is known to be injective on $\mathbb R$. – A.G. Jun 19 '21 at 18:15
@A.G. as Mathematica knows – mikado Jun 19 '21 at 19:02

3 Answers3