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MathGroup Archive 2006

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Re: Help with Identities

  • To: mathgroup at smc.vnet.net
  • Subject: [mg65588] Re: Help with Identities
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Mon, 10 Apr 2006 02:31:08 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <e1ahdv$2j1$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Sven C. Koehler wrote:
> Hello!
> 
> As an occasional mathematican, I sometimes forget that i.e.
> 
> Log[x/y] is very similar to Log[x] / Log[y]
> 
> Is there some way in Mathematica to see how an mathematical
> expression could look like alternatively?  (Something like the  opposite
> of Simplify.)
> 
> And then I wonder why 
> 
> Log[x/y] === Log[x] - Log[y]
> 
> is False.  Can I instruct Mathematica to explain why this is False?
> 
> Best wishes,
> 
> Sven
> 
Hi Sven,

According to _The Mathematica Book_, 5th Ed., "You should typically use 
=== when you want to test the structure of an expression, and == if you 
want to test mathematical equality (Section 2.6.8)."

However,

In[1]:=
Log[x/y] == Log[x] - Log[y]

returns the expression unevaluated

Out[1]=
     x
Log[-] == Log[x] - Log[y]
     y

Mathematica works by default with complex numbers; therefore we must use 
some assumptions that indicates that x and y are positive real variables 
as in

In[2]:=
Simplify[Log[x/y] == Log[x] - Log[y],
   Assumptions -> x > 0 && y > 0]

Out[2]=
True

or we can use *PowerExpand* that makes this kind of assumptions for us

In[3]:=
PowerExpand[Log[x/y] == Log[x] - Log[y]]

Out[3]=
True

Best regards,
Jean-Marc


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