Talk:Common logarithm

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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)

1 More on Mantissas?
2 explanation
3 Originator
4 Algorithm for log10(x)
5 Further notation
6 Hilarious
7 Gauss Quote
8 The approximation
9 Introduction

[edit] 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.

[edit] 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)

[edit] Originator

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

[edit] 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)

[edit] 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)

[edit] 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.

[edit] 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)

[edit] 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)

[edit] 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)

Talk:Common logarithm

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Contents

[edit] More on Mantissas?

[edit] explanation

[edit] Originator

[edit] Algorithm for log10(x)

[edit] Further notation

[edit] Hilarious

[edit] Gauss Quote

[edit] The approximation

[edit] Introduction

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