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All About Logarithms

- A Practical and Most Useful
- Guide to Logarithms
- by
- Mr. Hansen

John Napier

John Napier, a 16th Century Scottish scholar,

contributed a host of mathematical discoveries.

John Napier (1550 1617)

He is credited with creating the first computing

machine, logarithms and was the first to describe

the systematic use of the decimal point. Other

contributions include a mnemonic for formulas

used in solving spherical triangles and two

formulas known as Napier's analogies.

- In computing tables, these large numbers may

again be made still larger by placing a period

after the number and adding ciphers. ... In

numbers distinguished thus by a period in their

midst, whatever is written after the period is a

fraction, the denominator of which is unity with

as many ciphers after it as there are figures

after the period.

Napier lived during a time when revolutionary

astronomical discoveries were being made.

Copernicus theory of the solar system was

published in 1543, and soon astronomers were

calculating planetary positions using his

ideas. But 16th century arithmetic was barely up

to the task and Napier became interested in this

problem.

Nicolaus Copernicus (1473-1543)

Even the most basic astronomical arithmetic

calculations are ponderous. Johannes Kepler

(1571-1630) filled nearly 1000 large pages with

dense arithmetic while discovering his laws of

planetary motion!

A typical page from one of Keplers notebooks

Johannes Kepler (1571-1630)

Napiers Bones

In 1617, the last year of his life, Napier

invented a tool called Napier's Bones which

reduces the effort it takes to multiply numbers.

Seeing there is nothing that is so troublesome

to mathematical practice, nor that doth more

molest and hinder calculators, than the

multiplications, divisions... I began therefore

to consider in my mind by what certain and ready

art I might remove those hindrances.

Logarithms Appear

- The first definition of the logarithm was

constructed by Napier and popularized by a

pamphlet published in 1614, two years before his

death. His goal reduce multiplication, division,

and root extraction to simple addition and

subtraction. - Napier defined the "logarithm" L of a number N

by - N107(1-10(-7))L
- This is written as NapLog(N) L or NL(N) L

- This definition leads to these remarkable

relations - sqrt(N_1 N_2) 107(1-10(-7))((L_1L_2)/2)
- 10(-7)N_1 N_2 107(1-10(-7))(L_1L_2)
- 107(N_1) / (N_2) 107(1-10(-7))(L_1-L_2)
- which give the identities
- NapLog(sqrt(N_1 N_2)) 1/2(NapLogN_1NapLogN_2)

- NapLog(10(-7) N_1N_2) NapLogN_1NapLogN_2
- NapLog(107(N_1)/(N_2)) NapLogN_1-NapLogN_2

How Logarithms Work

- Logarithms are based on exponential functions.
- Common logs are based on ten raised to a power
- x 10y
- Natural logs, which are based on the number e

raised to a - power, are used mostly in higher and theoretical
- mathematics
- x ey
- Either of these functions can be graphed in the

normal way and produce the typical exponential

curve. Notice how x and y are interchanged in

these expressions.

- Logarithmic Notation
- For logarithmic functions we use the notation
- loga(x) or logax
- This is read log, base a, of x. Thus,
- y logax means x ay
- And so a logarithm is simply an exponent of some

base.

Inverse Relations and Functions

- We show an inverse function using the notation

f(x)-1. - A function is inverted by interchanging the x

and y values, then resolving for y. - Inverting a function reflects it across the

line - x y.

Logarithmic Function FAQs

- Logarithms are a mathematical tool originally

invented to reduce arithmetic computations. - Multiplication and division are reduced to simple

addition and subtraction. - Exponentiation and root operations are reduced

more simple exponent multiplication or division. - Changing the base of numbers is simplified.
- Scientific and graphing calculators provide

logarithm functions for base 10 (common) and base

e (natural) logs. Both log types can be used for

ordinary calculations.

Exponential Logarithmic Functions

- Exponential functions always have the variable in

the exponent - f(x) 2x is an exponential function
- f(x) x2 is not an exponential function
- Definition The function f(x) ax, where a is a

positive number constant other than 1, is called

an exponential function, base a.

Graphing Exponential Functions and Logs

- Lets graph y log3x
- Observation because x and y can be interchanged

in this equation, the graph of x 3y is a

reflection of y 3x across the line y x.

Since a0 (altgt0) 1, the graph of y logax, for

any a, has the intercept (1,0). As can be seen

from this function, the domain of x is all

positive real numbers, and the range is all real

numbers.

Laws of Exponents and Logarithms

- Because logarithms are exponents, the laws of

exponents apply to - all logarithmic operations. These laws include
- Law for Multiplication
- bx by bx y
- Law for Division
- bx / by bx y
- Law for Power of a Power
- (bx) y bx y
- Law for Negative Exponents
- b-x 1 / bx

Exponential and Logarithmic Relationships

- Conversion between log and exponential forms is

often a - convenient way to solve problems.
- Because x ay and y log ax are equivalent,

then - 2x 8 is the same as x log28
- Log problems are solved the same way
- log2x -3, the equivalent of 2-3
- And by the laws of exponents, we obtain
- x 1/23 or x 1 / 8

Exponents Characteristic Mantissa

- The exponent of a number N consists of an integer

or characteristic and a mantissa that follows the

decimal point. - The characteristic is determined by the number of

places the decimal point is moved from its

position when N is written in scientific

notation. - The mantissa of an exponent is a non-ending

decimal fraction following the characteristic.

This number is most often found in a table of

logarithms. - Tables of logarithms usually give mantissa values

in 4 or 5 decimal places. The user must manually

calculate the characteristic.

Properties of Logarithmic Functions

- We always assume that a is positive (ltgt 1), and

is a constant so - it can serve as a logarithm base. This is a proof

of the first - theorem of logarithms
- Proof For any positive numbers x y, loga (xy)

loga x loga y - Let b loga x and c loga y. This is

equivalent to - x ab and y ac
- Now multiply x and y
- xy abac, or, by a law of exponents, ab

c - As a logarithmic statement we can now write
- loga(xy) b c
- And replacing the values of b and c, our

solution is - loga (xy) loga x loga y

Additional Log Computations

- Other log theorems show these relationships
- logaxp p loga x - raise a number to a

power by multiplying the log of the number

take the root of a number by dividing - the log of that number.
- loga x / y loga x - loga y - divide two

numbers by - subtracting the logs of the two
- numbers.
- logb N loga N / loga b - change the base of

number N by dividing its log by the log of the

new base.

Using Logarithms

- Logarithm calculations produce answers as an

exponent. - To find the actual numeric solution of the

calculation, the antilogarithm of the result

must be found. - When using a calculator, this is done by raising

the base number to the power of that exponent. - Example Using common logs, find the value of

373. - Step 1 Using a calculator, find the common

logarithm of 37 (1.58201724), then multiply that

by 3, (4.704605172). - Step 2 Use a calculator to find the value of
- 104.704605172. The answer is 50653.

Log Calculations Using a Log Table

- The Problem multiply 37 by 143 using the handout

log tables. This table, originally produced in

1939, is accurate to four decimal places. - Determine the characteristic of these numbers
- 37 3.7 101 the characteristic is 1.
- 143 1.43 102 the characteristic is 2.
- Using the table, find the number 37 in the

left-most column, and read its value in the

second (0 column) as .5051. The log of 37 is thus

1.5682. - You can check this with your calculator by

computing 101.5682 which is 36.99985312.

- Next find the log of 143. For this mantissa, we

use the log of 14.3 look this up under the

number 14, then go to column 3. This mantissa is

.1553. (Notice how the first digit of the

mantissa is only printed in column 0.) The

characteristic is now added to the mantissa,

making the log of 143 to be 2.1553. - Now add the two logs together 1.582 2.1553

3.7235. This is the logarithm of our answer. - The answer, (103.7235 or 5291 by calculator), is

discovered by finding the number from the table

that is closest to .7235, our mantissa, then

multiplying it by 10 raised to the power of the

characteristic.

- Our mantissa is found in column 9 under the

number 52. The resulting number, then, is 5.29,

remembering that our mantissa is less than 1. - The final answer is calculated as 5.29 times 10

raised to the power of the characteristic 3, or

1000. The result, 5290, is one less than the

actual answer of 5291, but is within 99.98 of

the actual value. - We will now repeat this multiplication, this time

using the log values produced by our calculator

rather than the tables.

Log Computations Using a Calculator

- The four-place log tables provide limited

accuracy compared to - that offered by todays scientific and graphic

calculators. - Henry Briggs (1560-1631) produced the most

accurate table of common logs for more than 300

years when in 1631 he published a book of 30,000

logarithms accurate to 14 decimal places. - In 1952 Professor Alexander J. Thompson published

his 20-figure log tables. This project was

started in 1924 to celebrate the tercentenary of

Briggs tables, but when finished, the tables

could have been generated in a matter of minutes

by newly developed computers. - The following exercise repeats the multiplication

just done, this time using a graphing calculator

that carries results to 14 decimal places with 2

digit exponents.

- On your calculator, start by pressing the LOG

button, then the number 37, followed by the

closing parenthesis. Press ENTER and observe the

results - LOG(37) 1.568201724
- Now do the same, this time with the number 143.
- LOG(143) 2.155336037
- Press the 2nd button followed by the 10x (LOG)

button, then enter the sum of these two numbers

(3.723537761). - The result, 5290.999994, is much closer to the

actual product of 37 and 143 (5291) than we

obtained using the four-place log tables. This

result is within 99.99999998 of the actual

value.

Conclusion

- Logarithms were originally devised to simplify

arithmetic. - Logarithms are simply exponents of some base

(usually base 10), and therefore follow all the

rules of exponents. - Logarithms can use any base, but only two bases

are normally used today Common Logs use a base

of 10, and Natural Logs use e as their base. - The modern calculator has nearly eliminated the

need or use of logarithms in ordinary

calculations. Theoretical math and physics,

however, frequently employ the use of Natural

Logs.

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