Logarithms self taught by peter h selby

In case you have forgotten what an "exponent" is, it is the small number placed to the right and slightly above another number to indicate the power to which the second number is to be r

Senior Research Engineer

General Dynamics/Astronautics

Formerly Flight Training Supervisor

Convair Division, San Diego

who taught me to

enjoy mathematics

LOGARITHMS SELF-TAUGHT

Copyright © 1964 by McGraw-Hili, Inc

All righh reserved, including the right to reproduce this book,

or any portion thereof, in any form, without permission by the Publisher

Printed in the United States of America

Library of Congress Catalog Card Number 64-16492

345678910 MUMU 7654321

ISBN 07-056068-4

www.pdfgrip.com

PREFACE

You will find the material in this book presented in a different way than you are accustomed to in an ordinary textbook The first thing you will notice is that you are given only a small amount of information at a time, then immediately asked an appropriate question to help you find out how much you have understood, just as an individual tutor would do The answer you select in each case determines the item of information you will be given next; hence your route through the course This

is a special kind of self-teaching technique known as grammed instruction" or "auto-instruction."

"pro-Your rate of progress-the time you spend and the amount of material you read-is, therefore, up to you You can go as rapidly

or as slowly as your aptitude and inclination permit

in this new way, you should find it an interesting way to learn

This book is intended primarily for the following two groups:

the practical techniques of solving mathematical problems which require, or can be simplified by the use of, logarithms (either numerical or trigonometric) and who are approaching this subject for the first time

high school or college (perhaps a long while ago), but who have had little occasion to use their knowledge since and now need a brief review of the subject in order to apply it in their work or to further study

www.pdfgrip.com

CONTENTS

Preface

Whom This Course Is For

What You Can Expect to Learn from This Course

What the Reader Needs to Know about This Book

Part I Logarithms of Numbe"

The Two Parts of a Log 19

Practice in Finding Logarithms 26

Use of the Log-Trig Tables 49

This specialized course of instruction is by no means

con-,-'epts underlying the use of logarithms, both numerical and trigonometric, and to assist you in learning how to apply these concepts in solving problems

Specifically, at the conclusion of this course, you should be generally familiar with the theory and use of common logarithms and the practical techniques of solving plane right triangles with the aid of a table of logarithms of the trigonometric functions You should also be able to demonstrate your knowledge by scor- ing at least 40 on the self-administered quiz found at the back of this book

It is assumed that the reader has had a first course in algebra and a course in trigonometry at some time in his educational career However, the author has endeavored to explain all

Glossary, page 66, for an explanation of any words or expressions that are not familiar to you )

You will not need to have studied trigonometry in order to understand the explanation concerning the theory and use of

www.pdfgrip.com

common logarithms However, if you wish to gain a working knowledge of how logarithms of the trigonometric functions are used to aid in the solution of problems in trigonometry, then you may wish to procure a copy of "Trigonometry Self-Taught," a companion volume which will supply the necessary background for this course, or to take one of the courses in trigonometry available through schools

is as much as you want (or need), practicing what this program will teach you should in time make you skillful in using logarithms

to solve practical problems in multiplication, division, raising numbers to higher powers, or extracting roots You also will develop a practical skill in solving plane right triangles through the use of logarithms

other applications and other kinds of logarithms-then by all means seek out a good teacher and a good standard textbook This program will at least have served to start you on a fascinating highway that you can follow as far as you care to travel

This is no ordinary book Although the pages are bered consecutively, they are not intended to be read consecu- tively

num-At the bottom of each page, you will be told which page to turn to next Follow these instructions and you will have no trouble staying on the intended path This will aid us in present- ing just the information you need to help you understand the subject matter

To begin, proceed to page 1

www.pdfgrip.com

Part I Logarithms of Numbers

Anyone who has ever had occasion to multiply 6.41 X 0.00127

X 6984.3 X 1.002, and then possibly divide the product by 974.2 X 0.0394, is in the market for logarithms (or "logs" as the name usually is abbreviated) whether he knows it or not They are of considerable aid in multiplying, dividing, and raising numbers to higher powers

As a word of encouragement, if you have not hitherto been

imagined difficulties, do not be afraid! There is nothing very mysterious about logs They are not a disease (not in the ordi- nary sense at least) and, in any case, not a contagious one Perhaps you have learned something about them already Let's see if you can pick the best answer to this question : Of what use are logarithms?

Logarithms aid in the multiplication and division

Logarithms can be used to add long sums of

I can't answer because I don't know what a

You're out of bounds

Apparently you didn't follow instructions Nowhere in this book are you directed to this page

directions at the bottom of each page in order to stay on the right path This book will not make much sense if the pages are read consecutively

Please return to page 1 and try again

www.pdfgrip.com

Read page 1 again; then choose the correct answer

Your answer: 9 is the base and 2 is the exponent

No, apparently you've missed the key point In describing exponents, we stated that they were located to the right and slightly above another number (the base) Probably you've confused the two in this case because the numbers we used, 2 and 9, are the same numbers we used in first discussing exponents

on page 5, but now we are using them in a different relationship!

Return to p age 5, look over the explanation again, and select the correct answer

www.pdfgrip.com

as being useful in mathematical problems-many of which occur

in our daily lives-where we are confronted with the rather tedious task of multi.plying and dividing long numbers

Actually, logarithms are nothing but exponents In case you have forgotten what an "exponent" is, it is the small number placed to the right and slightly above another number to indicate the power to which the second number is to be raised; that is, the number of times the second number is to be multiplied by itself

squared"), 2 is the exponent and 9 is called the "base." The term 92

may also be read "nine to the second power." Similarly,

43 (commonly read "four cubed") may be read as "four to the third power." We could, in fact, have the number 4 raised to

9 is the base and 2 is the exponent

2 is the base and 9 is the exponent

I'm still not clear which is which

PAGE 4 PAGE 8 PAGE 6

Your answer: I'm still not clear which is which

numbers involved here-one the base and the other the exponent

By our definition the exponent is the one slightly raised and to the right of the other number (the base)

With this in mind, read the question on page 5 again and pick the correct answer this time

www.pdfgrip.com

FROM PAGE 1 7

Your answer: I can't answer because I don't know what a

logarithm is

right-about the state of your knowledge, at least But come now You don't really have to have any great knowledge of logarithms to answer that question, do you? All you need to know is cont:lined in the opening paragraph of this subject Turn back to page 1, read the opening paragraph again, then select a better answer

Your answer: 2 is the base and 9 is the exponent

Good! You weren't fooled by the fact that we used the same numbers as before but interchanged their positions

Now in discussing exponents thus far, we have selected our base numbers more or less at random Suppose for a moment that, instead of choosing a base at random (such as 4 in the ex- pression 43 or 9 in the expression 92

), we choose the specific base 10 Then, raising 10 to various powers we get

101 = 10 1()2 = 100

103 = 1,000

104 = 10,000

105 = 100,000

and so on

The power indicators I, 2, 3, 4, 5, etc., are exponents And

once we select a specific base for raising to various powers, we

we get a system known as "common log.s." Ihis is the sy~

used most commonly (hence the name) and the only one with

Considering the foregoing, then, we can define a logarithm

as follows: The logarithm of a number is the power to wbicluz

the exponents in the above example may be called logs

us

Pick the right answer below, and let's see if you're still with

Common logs are simply powers of the base 10 PAGE 10

I still don't understand the definition of a log PAGE 9

Common logs is the only system of logs there is PAGE 12

www.pdfgrip.com

FROM PAGE 8 9

Your answer: I still don't understand the definition of a log

You probably understand more than you realize, but let's take

The logarithm of a number is the power to which a given base must be raised in order to produce that number

Let's take the number 10,000 To what power would you have to raise 10 in order to produce this number? You'd have

to raise it to the fourth power, wouldn't you? In this case, 10,000

is the number we wish to produce, 10 is the base, and 4 is the power to which the base must be raised in order to produce (get) 10,000 That's clear enough, isn't it?

Now, re-read the explanation on page 8 and select the rect answer

cor-Your answer: Common logs are simply powers of the

The log of 143 would be greater than 2 but less

www.pdfgrip.com

FROM PAGE 10 11

Your answer: The log of 143 will be greater than 3

than 1,000 In order for the log of a number to be greater than

Return to page 10 and try again

Your answer: Common logs is the only system of logs

there is

No; if you believe this, then we've left you with the wrong impression In selecting the base 10 for our system of logs, we

didn't say this was the only possible system What we said was

that selecting a specific base resulted in the production of a (just

a different system of logs The system based on 10 just happens

to be the one most convenient for most of our calculating because

it ties in very happily with our decimal numbering system But the selection of any consistent base, though it might result in a rather complicated system, will yield a family of logarithms

With this thought in mind, return to page 8 and choose a better answer

www.pdfgrip.com

Appendix A LOGARITHMS OF NUMBERS

Referring to the first page of the log tables (see preceding

margin of the page; this is where we find the first two digits of our

of four- and five-digit numbers also, but they will not be necessary for our purposes.)

Looking down the number column at the left, we come to 14,

in which to find the log of the number 143

the top of the page, then read down to the value shown posite 14

of 143 that we have been seeking

What did you get?

Write your answer in the space below, then tum to page 16

-www.pdfgrip.com

FROM PAGE 10 15

Your answer: I have no idea what it would be

to get some idea of how large a log it would take to produce any given number, all we have to do is say to ourselves, "To what

power-or approximately what power-would I have to raise the

base 10 to produce the number I want?"

answer would be simple: we'd have to raise 10 to the second

number 1,000, the answer would be just as obvious: we'd have

to raise 10 to the third power

Now think for a moment about where the number 143 lies with respect to 100 and 1,000, and the logs (powers) 2 and 3 reqm:ed to produce these two numbers, and then select the cor- rect answer on page 10

Answer: 0.15534

Did you get it right? Remember that because the tabulated

under-stood before every value found in the table; hence, we must insert

it when writing down these values And why are they decimal fractions?

Well, let's take the value just obtained from the table The

order to obtain the complete logarithm Therefore,

the base 10 must be raised to the second power to get 100, and

What would be the complete log of the number 14,300?

-Turn to page 19 for the correct answer

www.pdfgrip.com

FROM PAGE 10

Your answer: The log of 143 would have been greater

than 2 but less than 3

17

Good, you've caught the idea nicely The log value of 143 will certainly be greater than 2, for the log of 100 is 2 and our

will just as certainly be less than 3, for the log of 1,000 is 3 and

plus an additional decimal fraction Just how much this decimal fraction is we can only determine by consulting a table of log- arithms, such as that found at the back of this book

Let's turn to this log table (see Appendix A, page 73) then,

log of our number

When you have located Appendix A, mark it with your finger while you turn to page 14 for an explanation of how the log tables are laid out

Your answer: The characteristic of a number is the nearest

lower power of 10 contained in that number

analyze just what it is the characteristic actually represents

In actual practice, the characteristic is determined by tion For instance, our first step in finding the log of 143 was to determine the characteristic And we did so by recognizing that the base 10 would have to be raised to a power greater than 2 but less than 3 in order to produce the number 143 Hence the characteristic in this instance was 2

inspec-This is a perfectly valid procedure for determining the acteristic However, a rule-of-thumb method that is customarily used is to count the number of digits (figures) to the left of the decimal point and subtract one from this amount The resulting number is the characteristic Thus, in the foregOing example, there are three digits to the left of the decimal point; hence the characteristic of the number 143 is 2 (three minus one)

char-Let's try a few others for practice What are the istics of the following numbers?

character-27 37.2

496 7498.23 6.4 - -

You will find the answers on page 23

www.pdfgrip.com

FROM PAGE 16 19

Answer: 4.15534

The only part to change would be the whole-number part of the log, wouldn't it? The decimal fraction part would remain the same as that for the log of 143, since the digits are the same All of this brings out a very important point, namely, that the log of a number is composed of two parts These two parts are given the names "characteristic" and "mantissa." Thus the com- ponent parts of the log of 143 would be analyzed as follows:

The characteristic indicates how many digits

The characteristic of a number is the nearest

Your answer: Every number has a different characteristic

Remember what we said about the number 143? We said that in order to get this number we would have to raise the base ten to

mantissa, of course, but the "second power" refers to our teristic 2

charac-Raising 10 to the second power gives us the number 100 and

a characteristic of 2; raising it to the third power gives us 1,000 and a characteristic of 3 There are a lot of numbers between 100 and 1,000, and everyone of them will have the characteristic 2 True, they will each have a different "and then some," but they will all have the same characteristic until the numbers reach 1,000, then they will have the characteristic 3 until they reach 10,000, etc

I'm sure you see by now that many numbers will have the

same characteristic, so return to page 19 and pick the correct answer

www.pdfgrip.com

FROM PAGE 19 21

Your answer: The characteristic indicates how many digits

there are in a number

No, it doesn't tell us how many digits there are in a number

whether it's a three-, four-, or five-digit number we're working with We'll discuss this relationship very shortly, but in the

When you have given the matter a little further thought, return to page 19 and choose the right answer

Answers: 1 2.00432

2 1.75282

3 3.38561

4 0.95:176

discover your error ( s )

Now, since this is so much fun, let's try finding the log of a number whose value is less than unity (one), such as, for instance,

up the number in our last example )

Examining this number to determine its characteristic, we

appears unnerving enough after being told to compute the acteristic from the number of digits to the left of the decimal point But even more unnerving is the appearance of a zero between the decimal point and the first figure to the right! What are we to do? What will the characteristic be?

char-The characteristic of 0.0975 will be 2, one less

www.pdfgrip.com

FROM PAGE 18 23

Answers: 1, 1, 2, 3, and 0

Did you get them all right? Don't let the numbers to the

right of the decimal point influence you Remember that the

the decimal point

From what we have discussed so far, I'm sure you will see that this business of the characteristic can work the other way

means we know the characteristic, since it is part of the log), then the characteristic will tell us where the decimal point should go

in the number

For example, the mantissa 0.15534 that we found for the number 143 actually could represent anyone of several numerical

to find the number (or antilog as it's also called) which it

table and emerge with the figures 143 However, these figures might represent any of the following numbers:

143.0 14.30 1.430 0.1430 0.01430 1,430 1,4,300

values was correct would be by reference to the characteristic Knowing the characteristic is 2 tells us there should be 3 (one

more than the characteristic) digits to the left of the decimal pOint, and accordingly we would place the decimal point after the

3 (in our number 143)

Your answer: The characteristic of 0.0975 will be -1

I'm afraid not Had the number been 0.975, we could have concluded that the characteristic was -1 (one less than 0, the number of digits to the left of the decimal point) However, the fact that there is a zero between the decimal point and the first digit to the right certainly would indicate a characteristic other than -1

You must be pretty close to the truth by now, so look again

at the answers on page 22 and see if you can't spot the right one this time

www.pdfgrip.com

Suppose we wish to find the log of the number 9.75 First,

we must establish the characteristic of the number Following the rule of subtracting one from the number of digits to the left

looking in the number column of our log table (see preceding page) , we find the number 97; opposite it, in the column headed

"5" (representing the third digit in our number), we find the mantissa 0.98900 Since the characteristic is 0, our final result is,

FROM PAGE 22 27

Your answer: The characteristic of 0.0975 will be 2, one

less than the number of digits

number of digits in the whole number that determines the

number 0.0975, obviously the rule we gave you won't be of much direct help

There is, however, a way of approaching the problem Think about it a bit and then see if you can select the correct answer on page 22

Your answer: True

correctly) Does this indicate a negative characteristic of -2?

Is the mantissa still correct for the digits 975? The answer is no

on both counts

Obviously that's not the way to proceed, and that isn't what's indicated So return to page 31 and choose the other answer

www.pdfgrip.com

FROM PAGE 22 29

Your answer: The characteristic of 0.0975 will be -2

Good! That's precisely what it will be How did you arrive

at your answer? By the process of elimination? Well, there's

right the first time, so much the better

Although our rule looks a little outmoded in the present

situation, it could be used to get the correct answer For

of digits to the left of the decimal point) And since there is a zero between the decimal pOint and first digit to the right, it would be logical to assume the characteristic to be -2, which it is

Now rwte this: The characteristic in this case is one greater ( numerically) than the number of zeros between the decimal

point and the first digit to the right This fact gives rise to other convenient rule-of-thumb:

an-The characteristic of a number whose value is less than one (unity) is minus, and it is one greater, numerically, than the number of zeros between the decimal point and the first digit to

the right of the decimal point

name the correct characteristic in each case

One of the most important facts to understand and remember

before Why is this so important? Primarily because it means they are governed by the same rules that apply to exponents; this tells us how to use them for multiplying, dividing, etc

To refresh your memory, in case you've forgotten what you learned in your first year of high school algebra, here are the main rules that apply to exponents (and therefore to logarithms) :

of the divisor from the exponent (log) of the dividend

(log) by the higher power

the number by the root

It's only fair to warn you, in case one of your friends is a professor of mathematics, that if you show him the above rules he probably will view them with some alarm For we have disrobed them of most of the formal verbal attire with which the profes- sional mathematician usually clothes them and garbed them instead in everyday terms

But of course this is our approach throughout this book, so perhaps he will understand and forgive our intentional over-sim-

subject

With this momentary digression, let's go on to page 35 where you will see how these rules are applied

www.pdfgrip.com

I hope you didn't get fooled by any of the zeros that

ap-peared after the first digit to the right of the decimal point Our rule tells us to count the number of zeros between the decimal point and the first numeral to the right and add one to it to get

how many or what kind of digits (including zeros) come after

the first digit to the right; they have nothing to do with ing the characteristic

determin-Now, let's get back to our problem of finding the log of 0.0975

We have concluded that its characteristic is -2, and since it contains the same digits as the number 9.75, it will have the same mantissa, namely, 0.98900 Now, it might seem logical to assume that the log would be written

However, written in this way the minus sign would indicate that

the entire log was negative, whereas it's only the characteristic

that is negative For this reason, the log is written