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Logarithms:

An Introduction to Their Use

by

Robert Kern Curtis

1970

I. Scientific Notation.

Scientists write a number like 8,732,000,000,000,000 as 8.732E15,
while the number 0.000000002413 is written 2.413E-9.
This method of writing numbers is called scientific notation.

A decimal point is placed after the first digit which is not a zero.
The number is then multiplied by a power of ten. The exponent of the
power can be obtained by counting the number of decimal places from
the imaginary decimal point (immediately to the right of the first
digit which is not a zero) to the actual decimal point.

If the direction is to the right (as in the value of x on a graph),
the exponent is positive; if it is to the left, the exponent
is negative. If the imaginary decimal point and the actual decimal
point coincide, the exponent is zero. (We recall that any number,
except zero, with a zero exponent is equal to one.)

Notice the following examples:

a.  196,000 = 1.96E5, since 1.96 x 100,000 = 196,000.

b.  0.0000017 = 1.7E-6, since 1.7 x 0.000001 = 0.0000017.

c.  2.71 = 2.71E0, since 2.71 x 1 = 2.71

EXERCISE.

Express in scientific notation:

1.  8070000     8.  25.17         15.  .00005       22.  .207

2.  .000807     9.  163000000     16.  738          23.  .0000219

3.  80.7       10.  2230          17.  26.8         24.  8830

4.  3.13       11. .00352         18.  1.87         25.  .716

5.  .0632      12. 61100000       19.  22200000     26.  23.4

6.  1230       13. .000000123     20.  72.8         27.  .0000002

7.  .00547     14. 8.21           21.  81000        28.  7000

Express in standard notation:

29.     8.234E6                    33.     5.912E-3

30.     1.414E2                    34.     6.899E-1

31.     1.0E4                      35.     1.0E-2

32.     1.73E3                     36.     2.54E0

II. Logarithms.

In one word, a logarithm is an exponent. In operations with
logarithms, we follow the laws of exponents.

Unless the numbers are very easy to handle, using logarithms will
to a power, or extracting a root of a number. (Logarithms are not
easily used to add or subtract numbers.)

The logarithm of a number is defined as the exponent of
the power to which a given value called the base must be raised to
yield the number.

Three (3) is the exponent of the power to which the base two (2) must
be raised to give the number eight (8). This can be written:

log    8 = 3.
2

We read this: The logarithm of 8 to the base 2 is 3.

To illustrate how logarithms save time, do the following problems
by arithmetic:

3                1/2
1.  16 x 128      2.  8192/256      3.  (16)       4.  (4096)

Compare your work with the following method which uses a table of powers
(logarithms) of two (2) and applies the laws for exponents.

In this table, we use two (2) as the base. The exponents (E) of the
base are the logarithms. The number (N) in the chart indicates the
value of the power of two (2) in that column.

LOGARITHMIC TABLE FOR THE BASE TWO (2)

N  1  2  4  8  16  32  64  128  256  512  1024  2048  4096  8192  16384  32768

E  0  1  2  3   4   5   6    7    8    9    10    11    12    13     14     15

4    7    4+7    11
1.  16 x 128 = 2  x 2  = 2    = 2   = 2048

13  8    13-8    5
2.  8192/256 = 2  /2  = 2     = 2  = 32

3     4 3    12
3.  (16)  = (2 )  = 2   = 4096

1/2     12 1/2    (12)(1/2)   6
4.  (4096)    = (2  )    = 2          =2  = 64

These problems could have been done mentally by using the chart and
following the laws of exponents.

EXERCISE.

Using the logarithmetic table for the base 2, read off the answer
for each of the following:

1/5                                1/2
1.  (32768)                   4.  128 x (1024)

4 1/3      2
2.  64 x 512                  5.  (1024 x (16384) )   /(256)  x 8192

2
3.  1282

II.I Logarithmic and Exponential Functions.
x
There are two ways in which the relationship    10  = y    can be
expressed:

x = log   y
10

When x depends for its value on the value of  y; x is called a logorithmic
function of y.

x
y = 10

Here y depends for its value on the value of x; y is called an exponential
function of x.

In the first expression, we consider y as the independent variable,
and x as the dependent variable. In the second, we shift the role
of dependence. It is clear that the logarithmic function and the
exponential function are inverse functions. Each is single-valued
in the case of real numbers once a definite value is assigned
to the independent variable, the value of the dependent variable is
determined.

In working a problem by logarithms, we deal with both the logarithmic
and exponential functions. We first find the logarithms of the given
numbers and then follow the laws of exponents to find the logarithm
of the answer. After finding the logarithm of the answer, we find
the number, called the antilogarithm, which has that logarithm.
This number is the answer to the problem.

Any positive number, except one, may be used as a base in work with
logarithms. We shall make use of common or Briggsian
logarithms named after Henry Briggs (1561--1613). This system uses
ten (10) as its base.

In`advanced mathematics, particularly calculus and physics, frequent
use is made of the natural or Napierian system of logarithms
named in honor of John Napier (1550--1617) who was the inventor
of logarithms. This system uses as its base e,  an irrational
number whose value is 2.718281828459045.... The base e is the limit

1/x
of the value of (1+x)     as x approaches zero.

When natural logarithms are used, ln is used as the symbol for natural
logarithm (except in calculus books where log is used for natural
logarithm) and log is used for common logarithm.

3
If we take the identity 1000 = 10 , we may separate each

part and give it a name:

1000 is the number

10 is the base

3 is the exponent.

We may write the identity in logarithmic form as log   1000 = 3.
10

We read this: The logarithm of 1000 to the base 10 is 3.

We call the first form exponential--because we are using exponents.
We call the second form logarithmetic--because we are using logarithms.

In the remainder of this work, we shall use the base 10 in all problems
involving logarithms, and so we shall follow the usual convention

by writing log in place of log   --the base 10 being understood.
10

We can see the difference between the exponential and the logarithmic
forms in the following  examples:

Exponential                          Logarithmic

0
10  = 1                                    log 1 = 0

1
10  = 10                                   log 10 = 1

2
10  = 100                                  log 100 = 2

-1
10   = 0.1                                 log 0.1 = -1

-2
10   = 0.01                                log 0.01 = -2

We see that this system gives us the logarithms of every number that
is an integral (whole) power of ten.

Verify the following:

a.    log 0.00000001 = -8   b.    log 1,000,000 = 6.

Suppose we take a number between 100 and 1000: e.g., Find log 478.

log 100 = 2;                 log 1000 = 3.

Hence, log 478 is between 2 and 3. A decimal fraction must be added
to the 2. We call 2 the characteristic of the logarithm of
the number. The decimal part we call the mantissa.

The characteristic is determined by inspection of the number
(by the position of the decimal point). The mantissa is
determined from the logarithmic tables. Modern scientific calculators
will read out both characteristic and mantissa together.

II.II The Characteristic.

We use scientific notation to find the characteristic. Imagine a decimal
point after the first digit which is not a zero. Count the number
of decimal places from this imaginary decimal point to the actual
decimal point. For example, if you go six places to the right, as
in determining the value of  x  on a graph, the characteristic
is +6. If you go to the left four places, as on a graph, the characteristic
is -4. If the imaginary and actual decimal points coincide, the
characteristic is zero.

All the mantissas in the tables are positive (but scientific calculators
sometimes give negative mantissas). To avoid difficulties with a negative
mantissa, when the characteristic is negative it is customary to add
10 - 10 to the characteristic. This enables us to write a logarithm
with a positive characteristic and positive mantissa minus a negative
characteristic, without changing the value of the logarithm. For example,

log 4.53 = 0.6561

and so we write:

log 0.453 = 0.6561 - 1

or, following the usual procedure:

log 0.453 = 9.6561 - 10

EXERCISE.

Give the characteristics of the logarithms of the following numbers:

1.    21.6        7.    0.00271      13.   14.7      19.   687000

2.   21600        8.       6260      14.   1853      20.    0.862

3.   0.216        9.       0.07      15.  0.00005    21.   0.000111

4.  0.00034      10.      3.852      16.   31.3      22.    78.3

5.  2.7          11.      247.5      17.    8.8      23.    8654

6.  683          12.      0.83       18.  0.0073     24.    1.234

II.III The Mantissa.

Note well: ONLY THE MANTISSA IS FOUND IN THE TABLES.

There is a decimal point understood before each mantissa in the
tables. The numbers in the column under the N indicates the first
two digits of the number for whose mantissa we are looking. The number
on the top of the page and to the right of the N corresponds to the
third digit of the number for whose mantissa we are looking.

The numbers in the body of the tables are the mantissas. By looking
at the tables and locating 47 in the column marked N and then moving
along that line until you reach the column under the 8 on top of the
page, you will find the mantissa for 478. We find that the mantissa
for 478 is 6794, hence:

log 478 = 2.6794.

Verify the following:

log 2.32 = 0.3655               log 69600 = 4.8426

log 87.9 = 1.9440               log 0.0673 = 8.8280 - 10

In each case, the characteristic is the number in front of the decimal
point; the mantissa is the number behind the decimal point.

Note well: MANTISSAS DO NOT CHANGE WHEN THE SIGNIFICANT DIGITS
ARE THE SAME.

If the number has only one significant digit, supply two zeros to
find its mantissa in the tables; if it has only two significant digits,
supply one zero.

To find the mantissa of the logarithm of 2, look for the mantissa
of 200 in the tables; to find the mantissa of the logarithm of 45,
look for the mantissa of 450.

In finding the logarithm of a number, the position of the decimal
point determines the characteristic. The mantissa is determined solely
by the significant digits. The numbers 2, 20, 2000, 0.2, 0.000002
all have the same mantissa.

Similarly, note:

log 278 = 2.4440

log 2780 = 3.4440

log 27800 = 4.4440.

EXERCISE.

Write the logarithms of the following numbers:

1.  23.1       7.  4.68       13.  14.0       19.  2.31

2.  0.231      8.  70         14.  0.997      20.   763

3.  99.7       9.  200        15.  0.00616    21.  0.00009

4.   1        10.  58.3       16.  8.03       22.  0.73

5.  6.08      11.  0.271      17.  111        23.  81.3

6.  684       12.  0.2        18.  0.2        24.  900000

III. Antilogarithms.

In work with logarithms there are two processes:

a. Obtaining the logarithm of a number, given the number.

b. Obtaining the number, given the logarithm.

The second process, which is almost always the last step in doing
a problem by logarithms is called finding the antilogarithm of the
logarithm. It is the reverse process of finding the logarithm. On
calculators, it is sometimes called the INV or inverse of
x
finding the  logarithm, or  10    for common logarithms
x
and    e     for natural logarithms.

The antilogarithm of the logarithm of a number may be
defined as the number which equals the base raised to the power of
the given logarithm.
1.8745
If  log 74.9 = 1.8745 then 10       = 74.9.

For example: if log x = 5.4800, find x.

First find the given mantissa 4800 in the tables. Of what number is
it the mantissa? We see that it is the mantissa of 302. The significant
digits are 302. Use the characteristic to locate the actual decimal
point. Starting from the imaginary decimal point, go five places to
the right to locate the actual decimal point. Thus x = 302,000.

Verify the following:

If log x = 7.2833 - 10 then x = 0.00192

If log x = 9.7782 - 10 then x = 0.6

If log x = 0.9036 then x = 8.01

If log x = 4.9542 then x = 90000

IV. Logarithms and the Laws of Exponents.

Recall these laws of exponents:

3   2     3+2    5
(x )(x ) = x    = x

To multiply, we add the exponents of like bases;

9  3    9-3    6
x /x  = x    = x

to divide like bases we subtract the exponent of the
divisor from the exponent of the dividend;

3 2     (3)(2)    6
(x )  = x        = x

to raise to a power, we multiply the exponent of the
base by the the power;

9 1/3    9/3    3
(x )    = x    = x

to extract a root, we divide the exponent of the base
by the index of the root.

FOLLOW THE SAME RULES IN USING LOGARITHMS. LOGARITHMS ARE EXPONENTS.

IV.I . Multiplication.

To multiply two numbers, add their logarithms, and find the antilogarithm
of the resulting sum.

IV.II. Division.

To divide numbers, subtract their logarithms, and then find the antilogarithm
of this difference. Subtract the logarithm of the divisor from the
logarithm of the dividend.

IV.III. Raising to a Power.

To raise to a power, multiply the logarithm of the number by the exponent
and then find the antilogarithm of this product.

IV.IV. Extracting a Root.

To extract a root, divide the logarithm of the number by the index
of the root, and then find the antilogarithm of this quotient.

V. Cologarithms.

The logarithm of a number is the exponent of the power to which a
given number called the base must be raised to given the number.

The antilogarithm of the logarithm of a number may be defined as the
number which equals the base raised to the power of the given logarithm.

The cologarithm of a number is the logarithm of the reciprocal
of the number.

The reciprocal of a number is that number which when multiplied by
the given number yields a product of one. For example: the reciprocal
of 7 is 1/7; the reciprocal of 3/4 is 4/3.

Note: colog x = log 1/x;     colog 1/2 = log 2;   colog 8 = log 1/8.

The cologarithm of a number is easily found by subtracting the logarithm
of the number from zero, or from 10 - 10 if using the tables.

VI. Mantissas on the Slide-rule.

The mantissa of the logarithm of a number can be read to three significant
figures (instead of the usual four) from the L scale of the slide-rule.
This is particularly useful in finding roots and in raising to powers
while using the slide-rule. Simply find the number on the C scale
and the mantissa of its logarithm will be under the hair-line
of the cursor on the L scale.

GENERAL EXERCISE.

Solve each of these problems using logarithms. Find all answers to
four significant digits.

2
1.    (1/2) (2.71) (47.1 )

2
2.    Find the value of K in the formula K = WV /2g when
W = 268, V = 1070, and g = 32.16.

3
3.    The formula for the volume of a sphere is V = (4/3)(pi)r .
Find the volume if (pi) = 3.142 and r = 7.78.

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