2.102

SECTION 2

Definition of Logarithm.

The

logarithm

x

of the number

N

to the base

b is the exponent of the

power to which

b must be raised to give

N

. That is,

log b N

x

or

b x N

The number

N is positive and

b

may be any positive number except 1.

Properties of

Logarithms

1. The logarithm of a product is equal to the sum of the logarithms of the factors; thus,

log b

M

·N

log b M

log b

N

2. The logarithm of a quotient is equal to the logarithm of the numerator

minus the

logarithm of

the denominator; thus,

log

M

log b M

log b

N

b

N

log

pq

1

log

M

b

M

b

q

Other properties of logarithms:

log

b b

1

log

pq

p

log b M

b

M p

q

log

b1

0

log

N

log a

N

· logb

a

log

a

N

b

log

a

b

log

b(

b N ) N

b

log b N

N

Systems of Logarithms.

There are two common systems of logarithms in use: (1) the

natural

(Napierian or hyperbolic) system which uses the base

e 2.71828(2)the. . . ;

common

(Brigg-

sian) system which uses the base 10.

We shall use the abbreviation

log N inlogthis10 Nsection.

Unless otherwise stated, tables of logarithms are always tables of common logarithms.

Characteristic of a Common Logarithm of a Number.

Every

real

positive

number

has a real

common logarithm such that if

a

b , log

a

Neitherlogb zero. nor any negative number has a

real logarithm.

A common logarithm, in general, consists of an integer, which is called the

characteristic,

and

a decimal (usually endless), which is called the

mantissa.

The characteristic of any number may be

determined from the following rules:

GENERAL INFORMATION, CONVERSION TABLES, AND MATHEMATICS

2.103

Rule I.

The characteristic of any number greater than 1 is one less than the number of digits

before the decimal point.

Rule II.*

The characteristic of a number less than 1 is found by subtracting from 9 the number

of ciphers between the decimal point and the first significant digit, and writing

10 after the result.

Thus the characteristic of log 936 is 2; the characteristic of log 9.36 is 0; of log 0.936 is 9

10;

of log 0.00936 is 7

10.

Mantissa

of a

Common Logarithm of a Number.

An important consequence of the use of base

10 is that

the

mantissa

of a number is independent of the

position of the decimal

point. Thus

Examples:

log 10.35 1.0149;antilog

0.065 1.16.

2.104

SECTION

2

TABLE 2.23

Derivatives and Differentiation

Rules for differentiation.

From Baumeister and Marks,

Standard Handbook for Mechanical Engineers,

7th ed., McGraw-Hill Book

Company, New York (1967); by permission.

To find the derivative of a given function at a given point: (1) If the function is given only by a curve,

measure graphically the slope of the tangent at the point in question; (2) if the function is given by a mathematical

expression, use the following rules for differentiation. These rules give, directly, the differential,

dy, in terms of

dx; to find the derivative,

dy/dx, divide through by dx.

Here

u , v , w

, .

.

.represents any functions of a variable

x , or may themselves be independent variables.

a

is a constant which does not change in values in the same discussion;

e

2.71828.

1.

d(a u ) du

2.

d(au ) a du

3.

d(u v w · · ·) du dv dw · · ·

4.

d(uv ) u dv v du

5.

d(uvw

. . .)

(uvw

. . .)

du

dv

dw · · ·

u v

w

6.

u

v du

u dv

7.

d(u

m

) mu

m

1

d

v 2

du

v

Thus,

d(u 2 ) 2u du ;d(u 3) 3u 2du ; etc.

8.

dp

du

9.

d

1

du

u

2 p

u

u

u 2

10.

d(e u ) eu du

11.

d(a u ) (ln

a )a u

du

12.

du

13.

du

du

dln

u

dlog 10

u

(log 10

e )

(0.4343 . . .

)

u

u

u

14.

dsin u

cosu du

15.

dcscu cot u

cscu du

16.

dcosu

sin u du

17.

dsecu

tan

u

secu du

18.

dtan

u

2

19.

dcot u

2

sec u du

cscu du

20.

1

du

21.

1

du

dsin

u

p

dcsc u

1

u 2

u pu 2

1

22.

1

du

23.

1

du

dcos

u

p

dsec

u

1 u 2

u pu 2 1

24.

dtan

1

u

du

25.

dcot 1 u

du

1 u 2

1

u 2

26.

dln sin

u cot u du

27.

dln tan

u

2 du

sin 2u

28.

dln cos u tan

u du

29.

dln cot u

2 du

sin 2u

30.

dsinh

u

cosh u du

31.

dcschu

cschu

coth u du

32.

dcosh u

sinh

u du

33.

dsech u

sech u

tanh u du

34.

dtanh

u

sech

2

u du

35.

dcoth u

2

cschu du

36.

1

du

37.

1

du

dsinh

u

p

dcsch

u

u 2 1

u pu 2 1

38.

dcosh 1

u

du

39.

dsech 1

u

du

p

u p

u 2 1

1

u 2

40.

dtanh

1

u

du

41.

dcoth 1

u

du

u 2

u 2

1

1

42.

d(u v ) (u v 1 )(u

ln

u dv v du)

GENERAL

INFORMATION,

CONVERSION TABLES,

AND

MATHEMATICS

2.105

TABLE 2.24

Integrals

From Baumeister and Marks,

Standard Handbook for Mechanical Engineers,

7th ed., McGraw-Hill Book

Company, New York (1967); by permission.

An integral of

f (x ) dxis any function whose differential is

f (x ) dx, and is denoted by

f (x ) dx. All the integrals

of

f (x ) dx are included in the expression

f (x ) dx C

, where

f(x ) dx is any particular integral, and

C is an

arbitrary constant. The process of finding (when possible) an integral of a given function consists in recognizing

by inspection a function which, when differentiated, will produce the given function; or in transforming the

given function into a form in which such recognition is easy. The most common integrable forms are collected

in the following brief table; for a more extended list, see Peirce,

Table of Integrals,

Ginn, or Dwight,

Table of

Integrals and

other Mathematical Data,

Macmillan.

General formulas

1.

a du a

du

au

C

2.

(u v ) dx

u dx

v dx

3.

u dv uv

v du

4.

f (x ) dx

f [F (y )]F (y ) dy, x F (y )

5.

dy

f(x , y ) dx

dx

f(x , y ) dy