Talk:Common logarithm

Untitled

Has anyone else heard of the approximation at the end of the article? My dad said they used to use it back in college when calculators didn't do logarithms. It would be interesting to know how far back it dates, and how long, if ever, it was in common use. CyborgTosser 09:55, 17 May 2004 (UTC)

More on Mantissas?

I have a rather old book with a "Table of Mantissas" in the back and I was wondering if perhaps someone could either explain more on how to use such a table in the article, or explain how to use mine in this discussion page. Mine has a table of numbers down the side, beginning with 10 and ending with 99 going by 1s. The top has numbers from 0 through 9 going by 1s also. As such, it looks like this:

N    0    1    2    3   ... 9
10 0000 0043 0086 0128  ...
11 0414 0453 0492 0531  ...
12 0792 0828 0864 0899  ...
13 1139 1173 1206 1239  ...
14 1461 1492 1523 1553  ...
15 1761 1790 1818 1847  ...
16 2041 2068 2095 2122  ...
17 2304 2330 2355 2380  ...
18 2553 2577 2601 2625  ...
19 2788 2810 2833 2856  ...
.    .    .    .    .
.    .    .    .    .
.    .    .    .    .
99

Anyway, I was just kind of hoping someone could explain it to me...

Suppose you want the base-10 logarithm of 1.52. You look at "15" (the first two digits) in the left-marginal column, then find "2" (the third digit) in the top margin, then look at the entry: 1818. Conclude that log₁₀ 1.52, rounded to four digits, is 0.1818. Then suppose you want the base-10 logarithm of 1520, which is the number that results from starting with 1.52 and pushing the decimal point three places to the right. That logarithm is log₁₀ 1520 = 3.1818 (approximately). The "3" is how many places the decimal point was pushed; that's why using 10 as the base of logarithms is convenient when working in a base-10 numeral system. 3 is the characteristic and 1818 is the mantissa. Michael Hardy 01:15, 11 July 2005 (UTC)

Thank you.

explanation

i like this article, but i got directed to it due to interest in why the calculator made base-10 logs obsolete. unfortunately, this article really doesn't address that question.

for example, it states, "Because of their utility in saving work in laborious calculations by hand on paper, tables of base-10 logarithms were found in appendices of many books." as a non-mathematician, i have no idea why base-10 logs "save work..." and therefore i can't understand the connection to calculators. i'd love to see this issue fleshed out a bit.

s (unsigned by User:24.89.202.141)

Does the explanation at logarithm#Easier computations help? Maybe there should be a mention here to look at the explanation there. Gene Nygaard 19:11, 10 October 2005 (UTC)

Originator

Should this article discuss who invented Common logarithms? I think it was Henry Briggs. --maru (talk) contribs 16:35, 26 January 2006 (UTC)

Algorithm for log10(x)

Not that anyone gives a damn but here is an algorithm for calculating log10(x)

Does anyone know who invented this algorithm? —Preceding unsigned comment added by 130.75.46.195 (talk) 14:16, 20 December 2007 (UTC)

#!/usr/bin/python

from __future__ import division

def log10(X):
  epsilon = 0.000000000001
  integer_value=0
  while X < 1:
    integer_value = integer_value - 1
    X = X * 10
  while X >= 10:
    integer_value = integer_value + 1
    X = X / 10
  decfrac = 0.0
  partial = 1/10
  X=X*X*X*X*X*X*X*X*X*X
  while partial > epsilon:
    while X >= 10:
      decfrac = decfrac + partial
      X = X / 10
    partial = partial / 10
    X=X*X*X*X*X*X*X*X*X*X
  return (integer_value + decfrac)

if __name__ == '__main__':
  value = 4.5
  print "      X  =",value
  print "LOG10(X) =",log10(value)

Further notation

My old mathematics teacher at school also led me to believe that while ln is a standard abbreviation of log_e, lg was an accepted abbreviation of log₁₀. Has anyone else seen lg used?

I've seen it, but if I came across it without some context, I would be uncertain which base was intended. I suspect some people use lg for the base-2 logarithm. Michael Hardy 21:52, 24 May 2006 (UTC)

Yes, in computer science lg generally denotes log base 2. —Steven G. Johnson 04:03, 25 May 2006 (UTC)

Hilarious

I have to nominate "Numbers are placed on slide rule scales at a distance proportional to their common logarithm. By mechanically adding the distance for 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale, one can quickly determine that 2 x 3 = 6." for funniest Serious Caption on Wikipedia.

Gauss Quote

“ Gauss was a prodigious mental calculator. Reputedly, when asked how he had been able to predict the trajectory of Ceres with such accuracy he replied, "I used logarithms." The questioner then wanted to know how he had been able to look up so many numbers from the tables so quickly. "Look them up?" Gauss responded. "Who needs to look them up? I just calculate them in my head!" ”

.

Can be put in the article.--Shahab 19:39, 22 June 2007 (UTC)

The approximation

I've just changed the approximation formula which contained a mistake. Repeatedly taking the square root eleven times is equivalent to raising to the power 1/2048, not 1/2047 as given in the formula. The factor 889 is roughly 2048/log(10) (which is actually 889.4). Curiously, 2047/log(10) is very nearly 889 (it's 889.001) but this is mere coincidence. I presume it has something to do with why the original author had given 2047 instead of 2048 in the first place though. The main source error is the approximation for the logarithm of a number close to 1 (i.e. log x = x - 1). Using 889 instead of 889.435... will make the approximation worse for all numbers less than 1 and some greater than 1 but after a certain point, it will counter the effect of the log x = x - 1 approximation. That's certainly not a good reason to use 889 specifically though: you could say the same about using 1000 or 3551. The only reason to use 889 instead of 889.435... is that it's easier to remember and might be accurate enough for the users needs.

After all that, although it's an interesting calculator trick I don't think it should be included in the article unless someone can reference it. —Preceding unsigned comment added by 86.153.11.148 (talk) 01:08, 4 November 2007 (UTC)

Introduction

Let's have the most useful information first, please. I'm not too familiar with line spacing in this Wiki; it would be nice to even things out a little bit: lowering "4)", for example. Thanks.

In mathematics, the logarithm of x to the base a is written as log_ax and is defined by the statement:

$If~x=a^{y},~~then,~y=log_{a}x$

The logarithm is the inverse operation of exponentiation and obeys four laws:

$For~n,~x,~y\in \mathbb {R} ,~~x>0~and~y>0,$

$1)~~log_{a}(xy)~=~log_{a}x~+~log_{a}y$

$2)~~log_{a}\left({x \over y}\right)~=~log_{a}x~-~log_{a}y$

$3)~~log_{a}x^{n}~=~nlog_{a}x$

$4)~~log_{a}x~=~log_{a}y~~if~and~only~if~x=y$

--Charlesrkiss (talk) 04:56, 23 February 2008 (UTC)

That information should be in the logarithm article, not in the introduction here, since it's not specific to the common logarithm. If a reader is uncertain what a logarithm is, he or she can click the logarithm link in the first sentence of this article. —Bkell (talk) 06:31, 23 February 2008 (UTC)

Thanks. The link wasn't highlighted before.--Charlesrkiss (talk) 15:08, 29 February 2008 (UTC)

Copy of comment left within the article

CAN SOMEONE PLEASE PUT UP SIMPLE INSTRUCTIONS ON HOW YOU WORK OUT THE LOG10 OF A NUMBER THANK YOU —Preceding unsigned comment added by 141.241.29.239 (talk • contribs)

Use in plotting

Even though this article is short on sourcing, adding new unstuffed stuff like the big section on use in plotting, complete with pseudocode, is not acceptable. If we can verify that this technique is indeed used, via a WP:RS, then no problem. Dicklyon (talk) 18:26, 20 October 2009 (UTC)

The use of logarithms is an elementary and standard technique used for generating scales in many graphing environments. It is a fundamental technique that is ubiquitous in the computing industry.

The primary importance, however, is the demonstration that the identification of mantissa and characteristic is not an archaic or obsolete exercise given the modern convenience of automated computations of the logarithm function. —Preceding unsigned comment added by 208.124.199.186 (talk) 16:10, 26 October 2009 (UTC)

If it's standard, it shouldn't be hard to find a source and make it verifiable, yes? Dicklyon (talk) 23:14, 26 October 2009 (UTC)

What's with the Python code?

Is this in any way helpful to the article? Does it have a reliable source, or just the wiki? I expect I'll remove that as well as the bit on plotting that is being thrashed around, unless someone shows a source where they can be verified and explain whey they're useful. Dicklyon (talk) 16:08, 22 October 2009 (UTC)

Table

I changed the table to read as follows:

number	mantissa (from the table)	characteristic	log10(n)
n		floor(log10(n))	log10(n) − characteristic
5 000 000	0.698970004336018...	6	6.698970004336018...
50	0.698970004336018...	1	1.698970004336018...
5	0.698970004336018...	0	0.698970004336018...
0.5	0.698970004336018...	−1	−0.301029995663981...
0.000 005	0.698970004336018...	−6	−5.301029995663981...

It does not make sense to put the base-10 logarithm column first. That's against the whole spirit of this concept. The reason base 10 is used is that you can get the mantissa out of the table and then use a bit of common sense to find the characteristic and hence the base 10 logarithm. If you've got a calculator and can get the base-10 logarithm without doing that, then what's the point? The article should communicate the idea. Michael Hardy (talk) 17:55, 26 October 2009 (UTC)

(Oh, and by the way, whoever recently arranged the table to show them in a different order used a hyphen instead of a minus sign.) Michael Hardy (talk) 17:56, 26 October 2009 (UTC)

The table IS NOT a table of logarithms - it is a chart showing the derivation of the mantissa and characteristic of a number given its base 10 logarithm. The derivations are given by the second line of headings below each of the headings annotating the chart.

as per the chart:

Common Logarithm, Characteristic and Mantissa of Powers of 10 for a Number

number	log10(n)	characteristic	mantissa	combined form
n (= 5 × 10i)		floor(log10(n))	log10(n) - characteristic
5 000 000	6.698 970...	6	0.698 970...	6.698 970...
50	1.698 970...	1	0.698 970...	1.698 970...
5	0.698 970...	0	0.698 970...	0.698 970...
0.5	−0.301 029...	−1	0.698 970...	1.698 970...
0.000 005	−5.301 029...	−6	0.698 970...	6.698 970...

Note that the mantissa is common to all of the 5×10i. A table of logarithms will have a single indexed entry for the same mantissa. In the example, 0.698 970 (004 336 018 ...) will be listed once indexed by 5, or perhaps by .5 or by 500 etc. .

The chart lists the same mantissa several times and if one wishes to think of it as a table of logarithms, then it is a table of only a single entry - that of 5 - so it is a table of a logarithm. Why is that? It is because all multiplicative powers of 10 as stated earlier in the article are derived from any logarithm by simply counting the number of positions of the most significant digit from the decimal point to ascertain the value of the characteristic but the mantissa does not change. A table of logarithms thus lists only mantissas and not characteristics, which are listed in this chart.

—Preceding unsigned comment added by 99.254.69.141 (talk) 21:00, 26 October 2009 (UTC)

I understand that this is not a table of logarithms, but I think a table of logarithms would better serve the puprposes of this article. Michael Hardy (talk) 18:52, 11 November 2009 (UTC)

lg x

"To mitigate this ambiguity the ISO specification is that log10(x) should be lg (x)."

Whaaaat???? I always thought lg x was log₂ x. If ISO really says log10 it's not much of an ambiguity mitigation.

66.127.55.192 (talk) 10:09, 17 February 2010 (UTC)

"Uses"?

The "Uses" section doesn't actually mention any uses. There should certainly be several. Michael Hardy (talk) 21:44, 19 February 2010 (UTC)

Sentence fragment makes meaning unclear

"Such a table of "common logarithms" giving the logarithm, often to 4 or 5 decimal places, of each number in the left-hand column, which ran from 1 to 10 by small increments, perhaps 0.01 or 0.001."

That's not a sentence, and I don't understand the article's topic enough to fix it. I understand English grammar and syntax quite well, but I don't understand math or this article's topic well enough to correct the problems in the sentence.

Here's my best guess, with my suggested change in Bold text:

Such a table of "common logarithms" showed the logarithm, often to 4 or 5 decimal places, of each number in the left-hand column, which ran from 1 to 10 by small increments, perhaps 0.01 or 0.001. — Preceding unsigned comment added by 131.96.118.157 (talk) 22:01, 8 July 2011 (UTC)

User:Michael Hardy wrote it as a sentence, with "gave", in the orignal lead in 2003, but then mangled it a few years later. And nobody noticed until now. Aren't we all flaky? Dicklyon (talk) 03:05, 9 July 2011 (UTC)

And he has restored it to "gave" after I poked him about it. Dicklyon (talk) 19:54, 9 July 2011 (UTC)

Question

Is log(x) = 10^x? 121.7.54.103 (talk) 09:47, 2 May 2013 (UTC)

Naming

The article gives a list of variously obscure names -- I just encountered "decadic" somewhere, and wanted to check it was the obvious meaning -- but surely it is vastly more common to call it a "logarithm base 10", or "base 10 logarithm", than any form other than "common"? Imaginatorium (talk) 05:07, 18 February 2016 (UTC)