Re: Log[ E^x ]
- To: mathgroup at yoda.physics.unc.edu
- Subject: Re: Log[ E^x ]
- From: villegas
- Date: Sat, 11 Dec 1993 21:09:01 -0600
Iain "D." Currie <iain at cara.ma.hw.ac.uk> asked: > I know x > 0. How can I make > > Log[ E^x ] > > evaluate to x?? > > Iain Currie The friendly rules of pre-calculus and calculus can usually be enforced by applying the PowerExpand operator to the expression. For your example: In[1]:= Log[E^x] //PowerExpand Out[1]= x This will ignore the question of whether 'x' falls in the image of the principal branch of the logarithm function. Since you know x does (the branch chosen by the built-in Log maps the complex plane, punctured at the origin, to the open-closed strip -Infinity < Re[z] < +Infinity && -Pi < Im[z] <= +Pi, which contains the positive real axis), this formula is valid. By the way, we can see which branch of the logarithm the built-in 'Log' is by using the PolarMap function in the package ProgrammingExamples`ComplexMap` : In[4]:= Needs["ProgrammingExamples`ComplexMap`"] In[5] := ? PolarMap PolarMap[f, {r0:0, r1, (dr)}, {phi0, phi1, (dphi)}, options...] plots the image of the polar coordinate lines under the function f. The default values of dr and dphi are chosen so that the number of lines is equal to the value of the option PlotPoints of Plot3D[]. The default for the phi range is {0, 2Pi}. Try plotting on a disk that has a small puncture in the center to avoid the origin; I chose E^-2 <= r <= E^3. And use a variety of theta intervals of length 2 Pi to see that the image is always contained in the strip from -Pi to +Pi. PolarMap[Log, {E^-2, E^3}, {0, 2 Pi}, PlotRange->All] PolarMap[Log, {E^-2, E^3}, {-3 Pi, -Pi}, PlotRange->All] PowerExpand will perform other simplifications that are valid if the variables are confined to certain subsets of the plane: (1) Log[x y] ===> Log[x] + Log[y] In[22]:= Log[x y] //PowerExpand Out[22]= Log[x] + Log[y] (2) Log[x^p] ===> p Log[x] In[23]:= Log[x^p] //PowerExpand Out[23]= p Log[x] (3) Sqrt[x y] ===> Sqrt[x] Sqrt[y] In[24]:= Sqrt[x y] //PowerExpand Out[24]= Sqrt[x] Sqrt[y] (4) (x y)^r ===> x^r y^r (generalization of 3) In[25]:= (x y)^r //PowerExpand r r Out[25]= x y Robby Villegas Technical Support Wolfram Research