Simplifying log expressions

[Tom Judson]

I would like to simplify the difference of two log expressions to show
that I get a constant, but

simplify((1/2)*log(2*t) - (1/2)*log(t))

just returns the expression. Does anyone know of an easy fix for
this? Preferably, I would like something that Calculus II students
could easily use.

Tom Judson

[Michael Orlitzky]

There's no global function for it, but what you want is to call
full_simplify() on the expression.

sage: f = (1/2)*log(2*t) - (1/2)*log(t)
sage: f.full_simplify()
1/2*log(2)

[JamesHDavenport]

Unfortunately, full_simplify has its own problems, notably with branch
cuts.
sage: f = (1/2)*log(2*t) + (1/2)*log(-t)
sage: f.full_simplify()
1/2*log(2)
Unfortunately, when t=-1, we have the sum of the logarithms of two
negative numbers, and therefore the imaginary part is 2i pi, not 0

[Michael Orlitzky]

On 01/13/2012 07:38 PM, JamesHDavenport wrote:
> Unfortunately, full_simplify has its own problems, notably with branch
> cuts.
> sage: f = (1/2)*log(2*t) + (1/2)*log(-t)
> sage: f.full_simplify()
> 1/2*log(2)

In my session, I had the difference of two logarithms. In yours above,
you've got the sum. Is that an actual sage session? I get something
different on 4.8.alpha6:

sage: f = (1/2)*log(2*t) + (1/2)*log(-t)
sage: f.full_simplify()

1/2*I*pi + 1/2*log(2) + log(t)

In the example below, with t=-1, both logs should have imaginary part pi
and real parts log(2) and zero respectively?

[JamesHDavenport]

I was using sagenb,org, so the output isn't actually a SAGE session,
but pasting from sagenb.org. It says it is 4.7.2.
Glad it's fixed. I guess I ought to download a 4.8 if I'm really going
to comment in more detail, given the apparent changes.

[William Stein]

On Sat, Jan 14, 2012 at 10:00 AM, JamesHDavenport
<J.H.Da...@bath.ac.uk> wrote:
> I was using sagenb,org, so the output isn't actually a SAGE session,
> but pasting from sagenb.org. It says it is 4.7.2.
> Glad it's fixed. I guess I ought to download a 4.8 if I'm really going
> to comment in more detail, given the apparent changes.

Quick hint: If you click the Text button in the upper right of the
notebook, you'll get something that looks like a normal Sage session.

>
> On Jan 14, 1:47 am, Michael Orlitzky <mich...@orlitzky.com> wrote:
>> On 01/13/2012 07:38 PM, JamesHDavenport wrote:
>>
>> > Unfortunately, full_simplify has its own problems, notably with branch
>> > cuts.
>> > sage: f = (1/2)*log(2*t) + (1/2)*log(-t)
>> > sage: f.full_simplify()
>> > 1/2*log(2)
>>
>> In my session, I had the difference of two logarithms. In yours above,
>> you've got the sum. Is that an actual sage session? I get something
>> different on 4.8.alpha6:
>>
>>    sage: f = (1/2)*log(2*t) + (1/2)*log(-t)
>>    sage: f.full_simplify()
>>    1/2*I*pi + 1/2*log(2) + log(t)
>>
>> In the example below, with t=-1, both logs should have imaginary part pi
>> and real parts log(2) and zero respectively?
>>
>> >> There's no global function for it, but what you want is to call
>> >> full_simplify() on the expression.
>>
>> >>    sage: f = (1/2)*log(2*t) - (1/2)*log(t)
>> >>    sage: f.full_simplify()
>> >>    1/2*log(2)
>

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--
William Stein
Professor of Mathematics
University of Washington
http://wstein.org

[JamesHDavenport]

Thanks. Given that, here's the sagenb (4.7.2) version, showing the bug
(wrong when t is negative real):
sage: t=var('t')
sage: f=(1/2)*log(2*t)+(1/2)*log(1/t)
sage: f.full_simplify()
1/2*log(2)
[William - this is actually quite a difficult area: see Beaumont et
al.,
Testing Elementary Function Identities Using CAD.
AAECC 18(2007) pp. 513-543.
http://www.springerlink.com/content/f1357425727485r0.]

On Jan 14, 7:22 pm, William Stein <wst...@gmail.com> wrote:
> On Sat, Jan 14, 2012 at 10:00 AM, JamesHDavenport
>

> > For more options, visit this group athttp://groups.google.com/group/sage-support

> > URL:http://www.sagemath.org
>
> --
> William Stein
> Professor of Mathematics
> University of Washingtonhttp://wstein.org
>

>  Screen Shot 2012-01-14 at 11.21.58 AM.png
> 73KViewDownload

[Michael Orlitzky]

On 01/15/12 13:18, JamesHDavenport wrote:
> Thanks. Given that, here's the sagenb (4.7.2) version, showing the bug
> (wrong when t is negative real):
> sage: t=var('t')
> sage: f=(1/2)*log(2*t)+(1/2)*log(1/t)
> sage: f.full_simplify()
> 1/2*log(2)

I created a ticket for this here:

http://trac.sagemath.org/sage_trac/ticket/12322

Thanks for the example!

[Greg Marks]

Dear Sage Developers:

There seems to be a similar issue in Sage Version 4.8:

   sage: a=log(6)/(1+log(2))
   sage: (6*exp(-a)-2^a).full_simplify()
   -(2^(log(3)/(log(2) + 1) + 1/(log(2) + 1))*3^(1/(log(2) + 1))*e^(log(2)^2/(log(2) + 1)) - 6)/(2^(1/(log(2) + 1))*3^(1/(log(2) + 1)))
   sage: (6*exp(-a)/2^a).simplify_full()
   2^(log(2/3)/(log(2) + 1))*3^(log(2)/(log(2) + 1))*e^(-log(2)^2/(log(2) + 1))

Sincerely,
Greg Marks

    ------------------------------------------------
   | Greg Marks                                     |
   | Department of Mathematics and Computer Science |
   | St. Louis University                           |
   | St. Louis, MO 63103-2007                       |
   | U.S.A.                                         |
   |                                                |
   | Phone: (314)977-7206                           |
   | Fax: (314)977-1452                             |
   | Web: http://math.slu.edu/~marks                |
    ------------------------------------------------

[Dima Pasechnik]

In gmane.comp.mathematics.sage.support, you wrote:
>
> Dear Sage Developers:
>
> There seems to be a similar issue in Sage Version 4.8:
>
> sage: a=log(6)/(1+log(2))
> sage: (6*exp(-a)-2^a).full_simplify()
> -(2^(log(3)/(log(2) + 1) + 1/(log(2) + 1))*3^(1/(log(2) +
> 1))*e^(log(2)^2/(log(2) + 1)) - 6)/(2^(1/(log(2) + 1))*3^(1/(log(2) + 1)))
> sage: (6*exp(-a)/2^a).simplify_full()
> 2^(log(2/3)/(log(2) + 1))*3^(log(2)/(log(2) + 1))*e^(-log(2)^2/(log(2) +
> 1))

Sage calls Maxima to do such kinds of computations. If one uses Maxima
on these expressions directly, it does not come up any better than that.
(Or perhaps one needs to know more about Maxima than I do).

Best,
Dmitrii

>
> Sincerely,
> Greg Marks
>