Log and You!David GonzalesBri JackLeAndra MillerKaity Cernik

History of Log!

• Created by John Napier in the 16th century • He developed them for astronomy research because

they needed to compute large numbers. • Used in such things as engineering, business, and

sciences such as biology, chemistry, and physics. • In Latin it is logarithmus by Napier • Logos in Greek means “reckoning, ratio” and arithmos

means “number” so it becomes “reckoning number” • Was worked on and made better by Henry Briggs (1561-

1630) who published tables for the logarithms

Napier!

• Napier was born in 1550 and died on April 4th, 1617 • He was born in Edinburgh, Scotland

• Died of gout • He didn’t go into school until he was 13 and dropped

out after a short period of time • He was believed to be a magician and used

necromancy (summoning the spirits of deceased) and alchemy (attempting to change base metals into gold)

• He was a religious man and was interested in the Book of Revelations

Calculators

• First Calculators were abathia• Earliest counting device was probably a form of tally stick• Fertile Crescent included clay shpaes, which represented counts of items,

probably livestock or grains, sealed in containers• Scottish mathematician and physicist John Napier noted multiplication

and division of numbers could be performed by addition and subtraction, respectively, of logarithms of those numbers

• First logarithmic tables Napier needed to perform many multiplications• William Oughtred and others developed the slide rule in the 1600s based

on the emerging work on logarithms by Napier• These common operations can be time-consuming and error-prone when

done on paper. More complex slide rules allow other calculations, suck as square roots, exponentials, logarithms, and trigonometric functions

Slide Rule

• In general, mathematical calculations are performed by aligning a mark on the sliding central strip with a mark on one of the fixed strips, and then observing the relative posistions of other marks on the strips. Numbers aligned with the marks give approximate value of the product, quotient, or other calculated result.

• The user dtermines the location of the decimal point in the result, based on mental estimaion. Scientific notation is used to track the decimal poin in more formal calculations. Addition and subtraction steps in a calculation are generally done mentally or on paper, not on the slide rule.

Log form

Exponential form

If the base and the exponent are given we compute a power

If the the exponent and the power are given we compute a root (or radical )

If the power and the base are given, we compute a logarithm

The logarithm of a number y with respect to a base b is the exponent to which we have to raise b to obtain y.

x = logby <---> bx = y

bx = yb=base

x=exponenty=power

Examples

Ex1)a.) 102 = 100 log10100 = 2

b.) 10-2 = 0.01 log100.01 = -2

c.) 100 = 1 log101 = 0

Ex2)a.) 23 = 8 log28 = 3

b.) 32 = 9 log39 = 2

c.) 251/2 = 5 log255 = 1/2

• Special Bases-Logarithms with respect to the base b=10 are called common logarithms, and logarithms with respect to the base e=2.71828... are called natural logarithms

• Common logarithms have a base of 10, and natural logarithms have a base of e.

Number Exponential Expression Logarithm

1000 103 3

100 102 2

10 101 1

1 100 0

1/10=.1 10-1 -1

1/100=.01 10-2 -2

1/1000=.001 10-3 -3

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stefan_waner/realworld/calctopic1/logs.html

Citations• Alfred, Peter. "What on Earth is a Logarithm? ." The University of Utah. N.p., 27061997. Web. 20 Apr 2010.

<http://www.math.utah.edu/~pa/math/log.html>. • "Logarithms." Logarithms. N.p., n.d. Web. 20 Apr 2010.

<http://www2.tech.purdue.edu/met/courses/met162/html/default_logarithms.html>. • "Ask Doctor Math ." The Math Forum@ Drexel. Drexel University, 27 10 1999. Web. 20 Apr 2010.

<http://mathforum.org/library/drmath/view/58047.html>. • "John Napier." Wikapedia. N.p., 15 4 2010. Web. 20 Apr 2010.

<http://en.wikipedia.org/wiki/John_Napier>. • "Henry Briggs." N.p., 07 1999. Web. 20 Apr 2010. <http://www-history.mcs.st-

and.ac.uk/~history/Mathematicians/Briggs.html>. • Marcus, Nancy. "Math Skills Review Logarithms." N.p., n.d. Web. 20 Apr 2010.

<http://www.sosmath.com/algebra/logs/log4/log4.html>. • Mashburn, Steve. "Documents and other resources to supplement the blog I Want to Teach Forever". April

18,2010 <http://sites.google.com/site/teachforever/>.• McAllen, D. "I Want to Teach Forever". April 18, 2010

<http://www.teachforever.com/2009/02/transforming-logarithmic-functions.htm>.• "Math Skills Review Logarithms." N.p., n.d. Web. 20 Apr 2010.

<http://www.chem.tamu.edu/class/fyp/mathrev/mr-log.html>. • Waner, Stephan. "Logarithms." N.p., Oct 1999. Web. 20 Apr 2010.

<http://people.hofstra.edu/stefan_waner/realworld/calctopic1/