Yes
any number <>0 can be expressed as number N =b ^n where b is the
base and n is the exponent.
(negative numbers will lead to complex n). The logarithm is the exponent.
we write (using base10) 1000=10^3 and 1/1000 =10^_3 so the log(10) of
1000 is 3 and for 1/1000 it is _3
Using logs we can add or subtract in place of multiplying and dividing
and before digital computers we used log10 based slide rules or tables .
The natural log with a base e=2.718.... has scientific interest as its
base turns up in math and in nature. ( try ^._1 ).
Don
On 17/12/2013 12:09 AM, Linda Alvord wrote:
Is this a true statement?
NB. Taking the logarithm of the reciprocal of a
NB. number changes the sign of all parts of the logarithm
NB. the original number.
N=: 4 2 1 0 0.5 0.25
^. N
1.38629 0.693147 0 __ _0.693147 _1.38629
]R=: % N
0.25 0.5 1 _ 2 4
^. R
_1.38629 _0.693147 0 _ 0.693147 1.38629
C=:1j1 _1j1 _1j_1 1j_1
^.C
0.346574j0.785398 0.346574j2.35619 0.346574j_2.35619 0.346574j_0.785398
]RC=: % C
0.5j_0.5 _0.5j_0.5 _0.5j0.5 0.5j0.5
^. RC
_0.346574j_0.785398 _0.346574j_2.35619 _0.346574j2.35619 _0.346574j0.785398
Linda
From: programming-boun...@forums.jsoftware.com
[mailto:programming-boun...@forums.jsoftware.com] On Behalf Of km
Sent: Monday, December 16, 2013 10:54 PM
To: programm...@jsoftware.com
Subject: Re: [Jprogramming] A complex question?
Off topic: I now know that taking the logarithm of the reciprocal of a
non-zero real number changes the sign of the real part of the logarithm of
the original number:
csrp NB. change sign of real part
(1r2 * -@(+ +) + (- +))"0
csrp 1j2 _3j_4
_1j2 3j_4
]rr =: 1 % 4 2 1 0.5 0.25 NB. non-zero reals
0.25 0.5 1 2 4
(^.@% -: csrp@^.) rr
1
(^.@% -: csrp@^.) -rr
1
--Kip
Sent from my iPad
On Dec 16, 2013, at 7:02 PM, "Dan Bron" <j...@bron.us> wrote:
Not sure. I suppose instead of
-@^.@(+/&.:*:)
we could write:
^.@%@(+/&.:*:)
or even:
^.@(+/&.:(*: :. (^&_0.5) ) )
But I'm not sure what this buys us.
-----Original Message-----
From: programming-boun...@forums.jsoftware.com
[mailto:programming-boun...@forums.jsoftware.com] On Behalf Of km
Sent: Monday, December 16, 2013 7:49 PM
To: programm...@jsoftware.com
Subject: Re: [Jprogramming] A complex question?
Dan, I haven't been following this thread, but know that minus the
logarithm of a positive number is the logarithm of the reciprocal. Is
that relevant?
^. 1r4 1r2 1 2 4
_1.38629 _0.693147 0 0.693147 1.38629
--Kip
Sent from my iPad
On Dec 16, 2013, at 3:42 PM, Dan Bron <j...@bron.us> wrote:
Raul wrote:
Is there a better way of doing this?
{: +. r.inv j./1 1
Marshall responded:
You can also use (+/&.:*:) in place of |@j./ , leaving you with
-@^.@(+/&.:*:)"1
Raul wrote:
Experimenting: the - is necessary and the ^. is not necessary.
(I do not get a hexagon without the minus, I do get a hexagon
without the ^.).
Immediately after writing this I realized the - is also unnecessary
- changing >./ to <./
What I love is that through some simple trig and a few experiments,
we got from {:@+.@(r.^:_1)@(j./) to +/&.:*: .
I suppose I find this particularly gratifying because I spent some
time trying to restate Raul's phrase in terms of simple arithmetic
operations, staying entirely in the real domain, and I eventually
reproduced Marshall's verb. Having spent so much time "simplifying",
when I got the final, irreducible result, I wondered at the need for
-@^. , and what its physical interpretation was.
Raul's original verb could be rendered in English as "the length
component of a polar coordinate (initially specified in Cartesian
terms)". Why should that length be expressed as the negative log of
a distance? Why not, as Don put it, "the raw distance"?
I know there are subtle and beautiful connections between the
trigonometric and exponential functions, and the e hidden in r. is
one expression of that. But I'm still not seeing the fundamental
physical
interpretation.
In other words, I wasn't surprised with the -@^. disappeared in
Raul's use case; I might've been more surprised if it'd persisted.
Anyone want to help me see it? Maybe the best illustration would be a
concrete use case where the -@^. isn't superfluous - one where where
it is not only necessary, but inevitable?
That is, a use case where -@^. has obvious physical interpretation,
when applied to the distance. Ideally one like Raul's, which
ultimately didn't involve complex numbers (i.e. a real-valued binary
[dyadic] operation on real numbers).
-Dan
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