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Log[Exp[x]]

• To: mathgroup at smc.vnet.net
• Subject: [mg5720] Log[Exp[x]]
• From: TTCJ34A at prodigy.com (DR JOHN C ERB)
• Date: Sat, 11 Jan 1997 14:29:12 -0500
• Sender: owner-wri-mathgroup at wolfram.com

>Dear friends, suppose x is a variable and when I type "Log[Exp[x]]" 
and then 
>enter, Mathematica gives me "Log[Exp[x]]" back. Could you please 
tell me how 
>to let Mathematica to return "Log[Exp[x]]" as x?
>Thank you very much.
>Cheng Li
>chengli at uab.edu

NOTE:

In: x==Exp[Log[x]]
Out: True

In: x==Log[Exp[x]]
Out: x == Log[Exp[x]]

WHY IS THE FIRST EVALUATED TO TRUE, WHILE THE
SECOND REMAINS UNEVALUATED?

FOR REAL NUMBERS, THE "LOG" AND "EXP" ARE MONOTONIC
INVERSES; THERE IS A ONE-TO-ONE CORRESPONDENCE.

FOR IMAGINARY NUMBERS, THIS IS NOT TRUE.
CONSIDER:

In: Exp[Pi I]
Out: -1

In: Exp[-Pi I]
Out: -1

BOTH OF THESE EVALUATE TO THE SAME VALUE: -1.
NOW WHEN WE TAKE THE LOG, WE HAVE:

In: Log[-1]
Out: I Pi

THE LOG[-1] EVALUATES TO (Pi I), BUT BOTH
EXP[Pi I] AND EXP[-Pi I] EVALUATE TO -1!

SO SINCE MATHEMATICA DOES NOT KNOW IF "X" IS REAL
OR IMAGINARY, IT DOES NOT KNOW WHICH WAY TO GO
FOR LOG[EXP[X]].

John C. Erb