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MathGroup Archive 1999

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Re: Solving equations involving Ln function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg19907] Re: Solving equations involving Ln function
  • From: adam.smith at hillsdale.edu
  • Date: Tue, 21 Sep 1999 02:22:44 -0400
  • References: <7s1pb3$9n9@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

You want to use PowerExpand[].  I have included the code below.  It is
worth noting that Simplify[] does not automatically use PowerExpand and
that is why the form Simplify[PowerExpand[g]] is used.  There can be
other times when it is necessary to perform something similar to
PowerExpand[] before simplifying.

Adam Smith

In[1]:=
g = Log[x]+delta*v == Log[(1-delta)*x/(1-delta^2)]+delta*Log[(1-delta)
*delta*x/(1-delta^2)]+delta^2*v
Out[1]=
                         2         (1 - delta) x
delta v + Log[x] == delta  v + Log[-------------] +
                                             2
                                    1 - delta

             (1 - delta) delta x
   delta Log[-------------------]
                          2
                 1 - delta
In[2]:=
PowerExpand[g]
Out[2]=
                         2
delta v + Log[x] == delta  v + Log[1 - delta] -

                2
   Log[1 - delta ] + Log[x] +

                                                     2
   delta (Log[1 - delta] + Log[delta] - Log[1 - delta ] +

      Log[x])
In[3]:=
newg = Simplify[PowerExpand[g]]
Out[3]=
               2
delta v - delta  v - (1 + delta) Log[1 - delta] -

                                   2
   delta Log[delta] + Log[1 - delta ] +

                      2
   delta Log[1 - delta ] - delta Log[x] == 0
In[4]:=
Solve[newg,x]
Out[4]=
                                2
{{x -> Power[E, (delta v - delta  v - Log[1 - delta] -

        delta Log[1 - delta] - delta Log[delta] +

                     2                       2
        Log[1 - delta ] + delta Log[1 - delta ])/delta]}}
In[5]:=
Simplify[%]
Out[5]=
                             2 1/delta  v - delta v
       (1 + delta) (1 - delta )        E
{{x -> --------------------------------------------}}
                            1/delta
                 (1 - delta)        delta


In article <7s1pb3$9n9 at smc.vnet.net>,
  Satyajit Bose <sgb2 at columbia.edu> wrote:
> Hello,
>
> I am trying to solve some equations involving the natural log
function.
> Mathematica 3.0 will not let me solve them since the relations are
> non-algebraic. Is there any way to restrict the domain to positive
reals
> or get Mathematica to use the exponential as an inverse, so that I can
> get a solution. I know that this can be done in another system,
presumably
> because it is less careful about atypical domain restrictions. Here is
> my input line and results in the kernel:
>
> In[1]:=
> Solve[Log[x]+delta*v==Log[(1-delta)*x/(1-delta^2)]+delta*Log[(1-delta)
*d
>
> elta*x/(1-delta^2)]+delta^2*v,x]
>
> Solve::tdep: The equations appear to involve transcendental functions
of
> the
>     variables in an essentially non-algebraic way.
>
> Out[1]= Solve[delta v + Log[x] ==
>
>            2         (1 - delta) x              (1 - delta) delta x
> >     delta  v + Log[-------------] + delta Log[-------------------],
x]
>
>                                2                             2
>                       1 - delta                     1 - delta
>
> I am hoping to get a solution that looks like this:
>
> x -> exp[v*(1-delta)]*(1+delta)^(1+1/delta)/delta)
>
> Thank you for all your  help.
>
> Sayajit Bose
> --
> Satyajit Bose
> Department of Economics                    New York, NY 10027
> Columbia University                        (212) 665-8208
> http://www.columbia.edu/~sgb2              sgb2 at columbia.edu
>
>


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