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MathGroup Archive 2004

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Re: Inverse of "PowerExpand"

  • To: mathgroup at smc.vnet.net
  • Subject: [mg51659] Re: Inverse of "PowerExpand"
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Wed, 27 Oct 2004 23:44:08 -0400 (EDT)
  • References: <clnf0f$ode$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Klaus,

let's take a slightly simpler example:

In[114]:=
v = PowerExpand[Log[f*f]]

Out[114]=
2*Log[f]

1st idea

In[115]:=
Simplify[v]

Out[115]=
2*Log[f]

... same as your example.

2nd idea

In[125]:=
ComplexExpand[v]

Out[125]=
2*I*Arg[f] + Log[f^2]

... oh, looks quite good, but the imaginary part is disturbing.
So let's tell Mathematica that f is >0 (and hence real)

In[123]:=
Simplify[ComplexExpand[v], f > 0]

Out[123]=
2*Log[f]

... still not ok!
But telling it in another way (Arg[f]==0) works:

In[124]:=
Simplify[ComplexExpand[v], Arg[f] == 0]

Out[124]=
Log[f^2]

Now back to your example:

In[1]:=
w = PowerExpand[Log[a*b*b/c]]

Out[1]=
Log[a] + 2*Log[b] - Log[c]

"Inversion" can be done by using

In[18]:=
Simplify[ComplexExpand[w], Arg[a] == 0 && Arg[b] == 0 &&
    Arg[c] == 0 && a > 0 && c > 0]

Out[18]=
Log[(a*b^2)/c]

But, when playing around with the conditions, trying to be logical (a>0 
should mean Arg[a]==0 etc.) and other, you can discover the most funny 
things. E.g. adding the condition b>0 yields

In[19]:=
Simplify[ComplexExpand[w], Arg[a] == 0 && Arg[b] == 0 &&
    Arg[c] == 0 && a > 0 && c > 0 && b > 0]

Out[19]=
2*Log[b] + Log[a/c]

Noch einen schönen Abend dabei...(have a nice evening experimenting...)
Wolfgang

Klaus G wrote:

> PowerExpand[Log[a*b*b/c]]
> 
> gives:
> 
> Log[a] + 2 Log[b] - Log[c]
> 
> Which Function will return the original "Log"?
> 
> Klaus G.
> 
> 


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