Definition:

The logarithmic function (with base b), written f (x)= logb (x), is the inverse of the exponential function bx.

 

Use for logarithmic functions…

·        Solving exponential functions

·        Turn multiplication into addition

 

History:            Before the invention of the calculators, first tables of logarithms and then slide rules based on logarithms were used as an aid to multiplication.

 

 

logb(bx) = blogbx = x.

In other words, y = logbx if and only if x = by.

Example:          log2 16 = log2 24 = 4

                        log5 (1/25) = log5 5-2 = -2

 

The Algebra of Logarithms

 

Here are the main rules for manipulating logarithms:

    • logbxy = logbx + logby.
    • logbxy = y logbx.
    • logb1=0.
    • logb0 is undefined (or -infinity).
    • logbx = (logbc)logcx.

The last rule shows that all logarithmic functions are the same except for a constant multiple.

 

 

Graph

 

 

Graphs of exponential functions are pretty easy to sketch, for they all go through the point (0,1) and are increasing everywhere if the base is larger than 1, and decreasing everywhere if the base is between 0 and 1. A logarithmic function and its inverse have graphs that are reflections of each other through the line y = x. The applet below shows the graphs that are reflections of each other through the line y=x.

 

 In y2 you can see how you have to type in log2x to get the right graph.

 

 

 

 

 

 

Properties of logarithmic functions

 

  • The domain of a logarithmic function is (0,inf). That means logbx is only defined for x>0. It is because the range of every exponential function is (0,inf.), and logarithmic functions are inverses of exponential functions.

 

  • Since all the graphs of exponential functions contain the point (0,1), the graphs of all logarithmic functions contain the point (1,0), the reflection of (0,1) in the line y = x. Logb 1 = 0 for all b.

  • If the base b is larger than 1, the function logb is increasing everywhere.. If the base a is between 0 and 1, then the function logb is increasing everywhere.

 

  • Because a1 = a, logarithmic identity1 above implies logb b = 1

 

 

The exponential function and the natural logarithm

 

 

The most important exponential function is ex, or also called the exponential function. It follows that its inverse, the logarithm with base e, is the most important of the logarithmic functions. The logarithm with base e is called the natural logarithm, and it is denoted ln().

 

  • The natural logarithm of x = ln x = loge b

The natural logarithm is the one, which has the nicest purely mathematical properties and is the one, which we use almost exclusively in calculus

 

  • The common logarithm of x = log x = log10 x

Many scientific calculators have buttons devoted to the natural logarithm and the logarithm base 10

 

 

The History of logarithms by Wikipedia

 

 

“Jaina mathematicians in ancient India first conceived of logarithms from around the 2nd century BC. By the 2nd century AD, they performed a number of operations using logarithmic functions to base 2, and by the 8th century, Virasena described logarithms to bases 2, 3 and 4. By the 13th century, logarithmic tables were produced by Musli mathematicians.

In the 17th century, Joost Bürgi, a Swiss clockmaker in the employ of the Duke of Hesse-Kassel, first discovered logarithms as a computational tool; however he did not publish his discovery until 1620. The method of logarithms was first publicly propounded in 1614, in a book entitled Mirifici Logarithmorum Canonis Descriptio, by John Napier, Baron of Merchiston in Scotland, four years after the publication of his memorable discovery. This method contributed to the advance of science, and especially of astronomy, by making some difficult calculations possible. Prior to the advent of calculators and computers, it was used constantly in surveying, navigation, and other branches of practical mathematics. It supplanted the more involved prosthaphaeresis, which relied on trigonometric identities, as a quick method of computing products. Besides their usefulness in computation, logarithms also fill an important place in the higher theoretical mathematics.

At first, Napier called logarithms "artificial numbers" and antilogarithms "natural numbers". Later, Napier formed the word logarithm, a portmanteau, to mean a number that indicates a ratio: λoγoς (logos) meaning ratio, and αριθμoς (arithmos) meaning number. Napier chose that because the difference of two logarithms determines the ratio of the numbers for which they stand, so that an arithmetic series of logarithms corresponds to a geometric series of numbers. The term antilogarithm was introduced in the late 17th century and, while never used extensively in mathematics, persisted in collections of tables until they fell into disuse.

Napier did not use a base as we now understand it, but his logarithms were, up to a scaling factor, effectively to base 1 / e. For interpolation purposes and ease of calculation, it is useful to make the ratio r in the geometric series close to 1. Napier chose r = 1 − 10 − 7 = 0.999999, and Bürgi chose r = 1 + 10 − 4 = 1.0001. Napier's original logarithms did not have log 1 = 0 but rather log 107 = 0. Thus if N is a number and L is its logarithm as calculated by Napier, N = 107(1 − 10 − 7)L. Since (1-10^{-7})^{10^7}is approximately 1 / e, L is approximately 107log1 / eN / 107. “

 

 

 

Graph

 

Let us look at a “real life” problem in which you need logarithmic functions:

Tim says that the temperature on the probe rises constantly when you use it for measuring you body temperature. Smart Smith says that it will create a logarithmic function…

 

He measured the temperature of his hand and got the following data, starting at 22.55 degrees Celsius.

 

Time (sec)

Temp (Celsius)

0

5

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

95

100

105

110

115

120

125

130

135

140

145

150

155

160

165

170

175

180

 

22.55

26.547

29.619

31.778

33.389

34.441

35.404

36.18

36.472

37

37

37.059

37.059

37.059

37.059

37.062

37.062

37.062

37.062

37.063

37.063

37.07

37.077

37.077

37.077

37.077

37.077

37.08

37.08

37.081

37.081

37.082

37.082

37.082

37.082

37.082

37.082

 

He did the Logistic regression and got the graph. He was right, it is a logarithmic function. Who would have thought so...?

 

In the time before Calcuators they had books full of common logarithms. This one it from the 20th century.

 

 

 

About me:

 

I am Konstantin Koerner and senior at Fayetteville High School. This Web page is made because my PreCalculus teacher told me so, it one of the last things I am doing in the High School. We need it for our Portfolio.

 

 

Links:

 

My PreCal teacher's web page

More about logarithms

More about logarithms at Wikipedia

More about logarithms

More about logarithms

More about logarithms

Logarithms

 

My e-mail