A History of Logarithms
MAT 320
Instructor: Dr Sunil Chebolu
By Maria Paduret
How LOGARITHMS appeared?
People didn’t know how to multiply or
divide big numbers. These calculations
were necessary not only in Commerce and
Business, but also in Astronomy,
Engineering, and Science. There were
“calculation centers” where people were
bringing their problem, paying and coming
back after a few days for the answer.
John Napier 1614
announced that from
that day on,
everybody will be
able to solve these
problems: the
multiplications and
divisions will be
replaced by simple
additions and
subtractions
Joost Burgi 1620
a clockmaker from
Switzerland published
almost the same thing
regarding logarithms.
He was also well
known as being a
maker of astronomical
instruments and
mathematician.
Objective
The objective of both men was to simplify
mathematical calculations. Napier’s approach
was algebraic while Burgi’s was geometric.
• “Mirifici
Logarithmorum
Canonis
Descriptio”
(Description of the wonderful canon of logarithms) by
Napier published in 1614
• There is evidence that Bürgi arrived at his invention as
early as 1588, six years before Napier began work on the
same idea. By delaying the publication of his work to
1620, Bürgi lost his claim for priority in historic
discovery.
HOW?
The logarithms have a very interesting
property:
log (ab) = log a + log b.
Therefore, with precise tables of logarithms,
if we want to multiply two numbers a and b,
we’ll have to look up into the table for log a
and for log b, we add the two values (S), and
then we go back into the table and find out
what number has S as a logarithm. This
number is the answer of ab.
Example
Multiply 15.27 by 48.54
log(15.27) = 1.1838 and log(48.54) = 1.6861
Add the logarithms
1.1838 + 1.6861 = 2.8699
It is possible to use the log tables backwards, but
most people would have turned to the next page
for the table of antilogarithms
The answer is 741.1
Logarithms
Definition:
A logarithm is the exponent or power to which a number
a, called the base, I raised to yield a specific number:
Let a > 0 be a real positive number, a nonequal to 1 .
Considering the equation:
ax = N, N > 0
(1)
(1) has a unique solution, and this solution is:
x = logaN
(2)
and we call it logarithm of N to the base a.
From (1) and (2) we get the following equality:
alogaN = N
If in (1) we make x = 1, we’ll obtain a1 = a hence
logaa = 1
Properties of Logarithms
•
loga(AB)=logaA + logaB
The property can be given also for n positive numbers
A1,A2,...,An:
• loga(A1A2…An) = logaA1 + logaA1 + logaA2 +…+ logaAn.
•
loga (A/B) = logaA- logaB
•
loga (1/B) = -logaB
•
logaAm=mlogaA,
where A is a positive number
loga (A1/n) = (logaA )/n
This is a particular case of the previous property, when
m=1/n.
•
logaA = logbA * logab
Why do you use logarithms?
• To find the number of payments on a loan or the time to reach an
investment goal
• To model many natural processes, particularly in living systems. We
perceive loudness of sound as the logarithm of the actual sound
intensity, and dB (decibels) are a logarithmic scale. We also perceive
brightness of light as the logarithm of the actual light energy, and star
magnitudes are measured on a logarithmic scale.
• To measure the pH or acidity of a chemical solution. The pH is the
negative logarithm of the concentration of free hydrogen ions.
• To measure earthquake intensity on the Richter scale.
• To analyze exponential processes. Because the log function is the
inverse of the exponential function, we often analyze an exponential
curve by means of logarithms.
• To solve some forms of area problems in calculus. (The area under
the curve 1/x, between x = 1 and x = A, equals lnA.)
References:
http://www.spiritus-temporis.com/logarithm/history.html
http://www.westcler.org/gh/Outtda/pdf_files/History_of_Logarithms.pdf
http://www.humboldt.edu/~mef2/Presentations/HSU%20Colloquia/colloq2_4_9
9.html
http://www.google.com/search?q=history+of+logarithms&hl=en&tbs=tl:1&tbo=
u&ei=u5gvTNiTCsrsnQfo9_zYAw&sa=X&oi=timeline_result&ct=title&res
num=11&ved=0CEkQ5wIwCg
http://www.math.umt.edu/TMME/vol5no2and3/TMME_vol5nos2and3_a14_pp.
337_344.pdf
http://www.stiintaazi.ro/index.php?option=com_content&view=article&id=3932
:istoria-matematicii-john-napier-scotianul-care-a-inventat-logaritmii-in1614&catid=171:istoria-matematicii&Itemid=124
http://en.wikipedia.org/wiki/Logarithm
http://books.google.com/books?id=8uYGAAAAYAAJ&printsec=frontcover&dq
=tables+of+logarithms&source=bl&ots=_Dsm62xbZB&sig=vdBYQEiE6eHdaHOEA9V3kjLU9I&hl=en&ei=jzIzTMzODYa0lQfwjom_Cw&sa=
X&oi=book_result&ct=result&resnum=1&ved=0CBEQ6AEwADgK#v=one
page&q&f=false
Questions?