simple math question

Could it possibly be said that a logarithm is a way to find
the "exponent" of a number??

I mean, in the most simple case...

??

J. Pehrson

--- In tuning-math@y..., "jpehrson2" <jpehrson@r...> wrote:
> Could it possibly be said that a logarithm is a way to find
> the "exponent" of a number??
>
> I mean, in the most simple case...
>
> ??
>
> J. Pehrson

Joe,

For any "base" B (such as 2, pi, e, 3, 5, 7, or good old 10),
the "logarithm" *is* the "exponent" TO WHICH THE BASE IS
TAKEN in order to equal some number N.

Examples:

In base 10, 10^2 = 10 x 10 = 100 = N
The exponent 2 is the power (thus the LOGARITHM)
that the BASE (of 10) is taken to in order to
equal the number N (100). So the logarithm
(to the base 10) of the number 100 equals 2.

In base 2, 2^3 = 2 x 2 x 2 = 8 = N
The exponent 3 is the power (thus the LOGARITHM)
that the BASE (of 2) is taken to in order to
equal the number N (8). So the logarithm
(to the base 2) of the number 8 equals 3.

J Gill :)

hi joe,

> From: jpehrson2 <jpehrson@rcn.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Saturday, February 02, 2002 10:56 AM
> Subject: [tuning-math] simple math question
>
>
> Could it possibly be said that a logarithm is a way to find
> the "exponent" of a number??
>
> I mean, in the most simple case...
>
> ??

the logarithm is t h e way to find the exponent

by definition, that's exactly its purpose

example:

the log_10 (the underscore is the ASCII way of writing a subscript)
-- which is read "log base 10" -- of 2, is ~0.30103

this simply means that 10^.30103 = ~2

of course, it's of utmost importance to know what the base of
the logarithm is, and people don't always indicate that explicitly

particularly in tuning math, logs are often taken to base 2
since 2 is the ratio of the "octave", and the author often
assumes that the reader will know that and assume 2 as the base

another thing to keep in mind is that logarithms can be taken
out to an arbitrary number of decimal places because the math
doesn't work out exactly as low-integer fractions ... hence my
use of the tilde(~) above to indicate approximations

however, i did notice something interesting early this morning
just before going to bed ...

there are some very good low-integer-ratio approximations to
the log_10 of the lowest primes

Examples:

log_10 ~fractional value
of: of logarithm

more accurate less accurate

2 3/10
3 10/21 = ~1/2
5 7/10
7 11/13 = ~5/6
11 25/24
13 39/35 = ~10/9
17 16/13 = ~11/9
19 23/18 = ~14/11 = ~5/4

this is useful because these simple fractions provide a very
easy way to work with approximate logarithms

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

> particularly in tuning math, logs are often taken to base 2
> since 2 is the ratio of the "octave", and the author often
> assumes that the reader will know that and assume 2 as the base

Another base often found in these parts is 2^(1/1200); the log base the 1200 root of two defines interval size in cents.

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:

/tuning-math/message/3185

>
> hi joe,
>
>
> > From: jpehrson2 <jpehrson@r...>
> > To: <tuning-math@y...>
> > Sent: Saturday, February 02, 2002 10:56 AM
> > Subject: [tuning-math] simple math question
> >
> >
> > Could it possibly be said that a logarithm is a way to find
> > the "exponent" of a number??
> >
> > I mean, in the most simple case...
> >
> > ??
>
>
>
> the logarithm is t h e way to find the exponent
>
> by definition, that's exactly its purpose
>

***Thanks, everybody, for the answers... It looks like I was "kinda
right" and "kinda wrong..."

It seems to hinge mostly on *definitions...*

Thanks!

JP