Logarithm table

Page from a four-digit log table from 1912

Logarithmic table from 1804

Logarithm table is called a tabular representation of the mantissas of logarithms . A more precise log table typically runs over several pages of a book. Logarithm tables have been an important arithmetic aid for centuries , especially in the natural and engineering sciences. Many calculations in school mathematics, e.g. B. the pulling of difficult roots, could only be done with their help. The invention and widespread use of pocket calculators and computers has made the use of log tables, similar to that of slide rules , practically completely superfluous within a few years.

The most common tables were the decadic logarithm (to base 10 ) with a resolution of 1.00 to 9.99.

history

For the history of logarithms, see the main article Logarithm: History .

With his work Mirifici Logarithmorum Canonis Descriptio, John Napier was the first to publish a logarithm table in 1614 and is considered its inventor. At first he was concerned with the simpler and more precise use of the trigonometric tables used at the time. In an Appendix Constructio , Napier considered taking a firm base, which his friend Briggs soon did.

Jost Bürgi was involved in the introduction and development of the decimal numbers, which were necessary for practical arithmetic, and calculated the first logarithmic table 1603-11 independently of Napier. Kepler urged him several times to publish it, but this did not happen until 1620 under Arithmetic and Geometric Progress Tabuln, according to Napier. As an employee of Johannes Kepler, he used the created log tables for astronomical calculations. These tables were purely numerical.

Henry Briggs introduced the 10 as a uniform basis in 1624 . He could no longer complete his table himself - here the logarithms of the numbers from 1 to 20,000 and from 90,000 to 100,000 were listed with 14 digits. It was published in full by the Dutch publishers Adriaan Vlacq and Ezechiel de Decker in 1627/28 in the Netherlands. The Vlacq tables contained a relatively small 603 errors. They completely displaced Napier's tablets and left no interest in Kepler's Chilias logarithmorum in 1624.

Tables were calculated using exponentiation. Only after the invention of infinitesimal calculus were more and more convergent series available for calculation.

With Nicolaus Mercator, it was possible to use series (1668 for ln (1 + x) ) for the calculation, but it took more than 100 years for Jurij Vega to publish his Thesaurus logarithmourum completus without errors in 1783 , which was the most famous table and for almost everyone lower figures formed the basis. Carl Bremiker improved the vegaschen food banks ( Vega-Bremiker ).

Use of log tables

Logarithm tables allow the multiplication and division of numbers to be reduced to the simpler addition and subtraction . Before there were mechanical or electrical calculators, log tables made arithmetic a lot easier. Logarithm tables were an indispensable companion in mathematics and physics classes in high school, for example.

The product of two numbers and is due to the law of logarithms ${\displaystyle a}$ ${\displaystyle b}$

${\displaystyle \log _{x}(a\cdot b)=\log _{x}a+\log _{x}b}$

calculated by looking up the logarithm of the base number and that of the base number in the table. The sum of the two logarithms is calculated and searched for in the table. The number resulting from this sum as a logarithm is then the product of and . ${\displaystyle a}$ ${\displaystyle x}$ ${\displaystyle b}$ ${\displaystyle x}$ ${\displaystyle a}$ ${\displaystyle b}$

With the help of a logarithm table, arithmetic operations can be traced back to the next simpler operation: multiplication to addition, division to subtraction, exponentiation to multiplication and square root (extraction of roots) to division. These returns are based on the following logarithmic laws:

${\displaystyle {\begin{aligned}\log _{x}(a\cdot b)&=\log _{x}a+\log _{x}b;&&a,b\in \mathbb {R} ^{+}\\\log _{x}\left({\frac {a}{b}}\right)&=\log _{x}a-\log _{x}b;&&a,b\in \mathbb {R} ^{+}\\\log _{x}(a^{b})&=b\cdot \log _{x}a;&&a\in \mathbb {R} ^{+},\ b\in \mathbb {R} \\\log _{x}{\sqrt[{b}]{a}}&={\frac {1}{b}}\cdot \log _{x}a;&&a\in \mathbb {R} ^{+},\ b\in \mathbb {R} \setminus \{0\}\end{aligned}}}$

Structure of a log table

The most common were three-, four-, and five-digit log tables. The greater the accuracy of a table, the greater its scope. Four-digit log tables were commonly used in schools until the 1970s.

Simple three-digit log tables are structured in such a way that the first two digits (i.e. 10 to 99) form the left edge of the table, while the third digit (0 to 9) serves as the column heading.

The number range from 1.00 to 9.99 is sufficient when using the logarithms for base 10. The logarithm of ten times, hundred times, etc. of a number can be calculated by modifying the integer part according to the number of places (number of places in front of the decimal point minus 1). See the Logarithmengesetz of multiplication . For example: The logarithm of the one-digit number 2 is about 0.30103; that of the two-digit number 20 is 1.30103; the logarithm of the three-digit number 200 is 2.30103 etc. For numbers smaller than 1, the following applies accordingly: and . ${\displaystyle \log _{a}(ax)=\log _{a}(a)+\log _{a}(x)=1+\log _{a}(x)}$ ${\displaystyle \log(0{,}2)\approx -1+0{,}30103=-0{,}69897}$ ${\displaystyle \log(0{,}02)\approx -2+0{,}30103=-1{,}69897}$

Logarithms for numbers with four valid digits can be determined by linear interpolation .

Since log tables were viewed as tools used every day, they were often enriched with additional information. For example , collections of formulas from geometry and trigonometry were included, collections of data, for example, about the bodies that make up our solar system , as well as life tables as examples of demographic data collections and much more

Creation of a log table

Logarithm tables were determined from value lists of the inverse function, the exponentiation, by interpolation.

PP tablets

Interpolation tables are included with the tables for linear interpolation. PP stands for partes proportionales and is a linear interpolation.

${\displaystyle x=\log _{b}a}$

Excerpt from the decadic (the base is 10) logarithm, number (the numerical value ) on the left and above, mantissa (here the decimal places are meant ) on the right for five-digit logarithms. The decimal places are divided into groups of twos and threes, the last three digits are on the right. In other tables, like here, for example, 82 is not repeated, but only written once in the column and only when it increases to 83 is written below it in the column: ${\displaystyle b}$ ${\displaystyle a}$ ${\displaystyle x}$

N	0	1	2	3	4th	5	6th	7th	8th	9
661	0.82020	027	033	040	046	053	060	066	073	079
662	0.82086	092	099	105	112	119	125	132	138	145

PP board:

PP	6th	7th
1	0.6	0.7
2	1.2	1.4
3	1.8	2.1
4th	2.4	2.8
5	3.0	3.5
6th	3.6	4.2
7th	4.2	4.9
8th	4.8	5.6
9	5.4	6.3

If you want to determine an interpolated mantissa for the number 66108, you have to add eight times the tenth of the table difference 7 (horizontal difference between the table values), i.e. 5.6, or 0.000056 and then rounded up would have m = 4.82026.

If you want to add another digit, you take parts of the table difference divided by 100 instead of 10. Only the last digit should be rounded. For the six-digit number N = 6613.78 in the first step 4.2 in the second 0.48 and then receives five-digit m = 82040 + 4.2 + 0.48 = 82045, i.e. 3.82045.

If you have for a four-digit number M = 82116 (3.82116) between M = 82112 and M = 82119, N must be between N = 6624 and N = 6625. The table difference is 7, the additional 4 of the mantissa is most likely to be found in the table, at 3.5 the number is 6624.5, if you round off 4.2, it would be 6624.6. 3.5 can be increased by 0.49 again, which means 0.07 in the table, so the number N is finally 6624 + 0.5 + 0.07 = 6624.57, which is rounded up to 6624.6. How to do the math with the calculator.

As you can see, tables are given for the differences 7 and 6, as both appear in the table, 027 to 033 are six, followed by seven again, 033 to 040.

Trivia

Tables of logarithms played a role in the discovery of Benford's law (actually by Simon Newcomb ). The side with the one as the leading digit is needed more often than the other digits and therefore wears out faster.

Known issues

• Vega-Bremiker , seven-digit logarithms and trigonometric functions, from 1795
• Wilhelm Jordan (geodesist) , logarithms and auxiliary tables
• FG Gauss : Five-digit complete logarithmic and trigonometric tables . (Over 100 editions since 1870).

Web links

Wiktionary: Logarithm table  - explanations of meanings, word origins, synonyms, translations

• Information on John Napier (University of St Andrews, Scotland )
• List of commonly used tables with table of contents and two illustrations of the book cover and one page.
• LOCOMAT, Loria Collection of Historical Tables

Individual evidence

1. ^ Athenaeum June 15, 1872. See also the Monthly Notices of the Royal Astronomical Society , May 1872.