Briggsian logarithms

Briggsian logarithms

The

Briggsian logarithm

of a positive number

a

is the

logarithm

of

a

in the base 10, i.e.

log

10

⁡

a

, nowadays denoted by

lg

⁡

a

(probably from the Latin “

logarithmus generalis

(

http://

planetmath

.org/TermsFromForeignLanguagesUsedInMathematics

)”).  The is due to Henry Briggs (1561–1630).  Before the electronic

calculators

and

computers

the tabulated values of logarithms were used for performing laborious numerical calculations (

multiplications

,

divisions

, powers, roots).  E.g. in the high schools of Finland, the use of logarithm tables was teached still in the begin of the 1970s.

There was several wide tables of Briggsian logarithms, e.g. the well-known five-place tables of Hoüel and Voellmy.  Since the logarithms of

rational numbers

are mostly

irrational

, the logarithms in the tables are in general approximate values.

Because

lg

⁡

10

⁢

a

=

lg

⁡

a

+

lg

⁡

10

=

lg

⁡

a

+

1

,

lg

⁡

a

10

=

lg

⁡

a

-

lg

⁡

10

=

lg

⁡

a

-

1

,

moving the

decimal point

one step to the right resp. to the left increases resp. decreases the Briggsian logarithm by the integer value 1; the decimals of the logarithm do not alter.  Thus the tables give only the decimals of the logarithms of

positive integers

.  For example, the table gives for the logarithm of 8322 only the five decimals 92023.  Since

lg

⁡

1

=

0

,

lg

⁡

10

=

1

and the

logarithm function

is

increasing

, we can infer that

lg

⁡

8322

≈

0.92023

+

3

lg

⁡

832.2

≈

0.92023

+

2

lg

⁡

83.22

≈

0.92023

+

1

lg

⁡

8.322

≈

0.92023

lg

⁡

0.8322

≈

0.92023

-

1

lg

⁡

0.08322

≈

0.92023

-

2

lg

⁡

0.008322

≈

0.92023

-

3

When one expresses logarithms of numbers as sum and

difference

in the way as above, the decimal part is called the

mantissa

and the

integer part

the

characteristic

of the logarithm.  A

positive

caracteristic is joined to the mantissa (e.g. 3.92023), but a negative characteristic is held apart (e.g.

0.92023

-

3

).

It’s clear that the mantissa of the logarithm of a number does not depend on the position of the decimal point in the number.  For obtaining the logarithm of a number from the table, one may drop the decimal point away and seek for the gotten

integer

the the mantissa of its logarithm.  Then one deduces the characteristic for the logarithm of the initial number.

Example.

Determine

63.873

3

as accurately as possible using five-place decadic logarithms.  We use the

log

⁡

a

3

=

log

⁡

(

a

1

3

)

=

1

3

⁢

log

⁡

a

.

We don’t find in the table so big numerus as 63873; therefore we take first the mantissa corresponding the numerus 6387, it is 0.80530.  The next mantissa, corresponding 6388, is 0.80536.  The difference of both mantissae is thus 6 units of the last decimal , and we could interpolate for getting the last mantissa decimal corresponding the numerus 63873.  For such interpolations there is on the same table page the auxiliary table P.P. (‘partes proportionales’) titled “6”; it gives for 3 the value 2 to be added to the last decimal .  So we have

lg

⁡

63.873

≈

1.80532

,

(1)

where the charactesistic 1 is infered from the fact that 63.873 is between 10 and 100.  Then the logarithm of the

cube root

is obtained by dividing (1) by 3:

lg

⁡

63.87

3

≈

0.60177

The mantissa digits 60177 are not found in the table, the most nearest are 60173 and 60184 which correspond the numeri 3997 and 3998.  The P.P. table titled “11” tells that we must join 4 to the end of 3997 (since

60177

-

60173

=

4

)

.  Thus we have got the result

63.873

3

≈

3.9974

.

This is the most accurate value with five places.

References

1

K. Väisälä:

Algebran oppi- ja esimerkkikirja II. Pitempi kurssi

.  Werner Söderström osakeyhtiö, Porvoo & Helsinki. Neljäs painos (1956).

2

G. J. Hoüel:

“

Tables de logarithmes à cinq décimales pour les nombres et les lignes trigonométriques…

”.  Gauthier-Villars, Paris. Sécond édition (1864).

3

E. Voellmy

: “

Fünfstellige Logarithmen und Zahlentafeln für die 90

o

-Teilung des rechten Winkels

”.  Orell Füssli Verlag, Zürich. Vierzehnte Auflage (1962).

Title

Briggsian logarithms

Canonical

name

BriggsianLogarithms

Date of creation

2013-03-22 16:39:37

Last modified on

2013-03-22 16:39:37

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

15

Author

pahio (2872)

Entry type

Topic

Classification

msc 26A09

Classification

msc 26-00

Classification

msc 65A05

Classification

msc 01-08

Synonym

decadic logarithms

Synonym

common logarithms

Synonym

lg

Related topic

RationalBriggsianLogarithmsOfIntegers

Related topic

LimitOfRealNumberSequence

Defines

mantissa

Defines

characteristic

Defines

numerus

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