appendix C

Calculations and Basic Logarithms

 

Calculation of the Mean¹

To determine the mean of a population or sample, all of the data are simply added together and divided by the number of pieces of data. For example, suppose that three tests of the resolving power of a film produced the following data: 59 lines/mm, 55 lines/mm, and 51 lines/mm. Using the formula

(where the symbol Ʃ means “sum of”) the steps are:

1.Add up all of the X values: (59 + 55 + 51 = 165).

2.Divide the total by the number of X values: (n) (165 3 = 55.0).

Notice that the mean is carried to one more decimal place than the original data. Thus, the mean resolving power for this experiment is 55.0 lines/mm.

Calculation of the Standard Deviation

Although the formulas for determining the standard deviation contain more terms, the calculations are relatively easy to do if taken step by step. To continue the example given above with resolving powers of 59, 55, and 51 lines/mm requires the use of one of the formulas for the sample standard deviation.

1. Determine the mean. Here ^— was found to be 55.0.
2. Determine the absolute difference (ignoring the sign) between each X value and the mean. In order, the differences are 4, 0, and 4.
3. Square each of the differences. The squares are 16, 0, and 16, respectively.
4. Add the squares. The total is 32. This is the numerator of the fraction under the radical sign.
5. Divide the total by the sample size minus 1: (n - 1 = 2) (32 ÷ 2 = 16).
6. Finally, find the square root of this value: >/16 = 4.0. The standard deviation is therefore 4.0 lines/mm, correctly taken to the same number of decimal places as the mean.

It is often convenient to set up the calculations in tabular form as shown:

(Step 1)	(Step 2)	(Step 3)
X	|X−X|	(X−X)2
59	55.0−59 = 4	16
55	55.0−55 = 0	0
51	55.0−41 = 4	16
ΣX = 165		(Step 4)
X = 165 ÷ 3		Σ(X-X)2 = 32
= 55.0
(Step 5)	(Step 6)
32 ÷ 2 = 16;	√16 = 4.0 = s.

Basic Logarithms

Both interval scales of numbers (1-2-3-4-5-6-etc.) and ratio scales of numbers (1-2-4-8-16-32-etc.) are used in photography. Interval scales, for example, are found in thermometers and rulers. Ratio scales, for example, are used for f-numbers (f/2-f/2.8-f/4-f/5.6-f/8) where every second number is doubled, and ISO values (100-125-160-200-250-320) where every third number is doubled. Even though certain quantitative relationships in photography conform naturally to the ratio-type scale, there are disadvantages in using ratio scales of numbers in calculating and graphing. The disadvantages include the difficulty of determining the midpoint or other subdivision between consecutive numbers, and the inconvenience of dealing with very large numbers.

Using the logarithms of numbers in place of the numbers eliminates disadvantages of ratio scales by converting the ratio scales to interval scales. Mathematical operations are reduced to a lower level when working with logs, so that multiplication is reduced to addition, division is reduced to subtraction, raising a number to a power is reduced to multiplication, and extracting a root is reduced to division.

Logarithms, or logs, are derived from a basic ratio series of numbers: 10, 100, 1000, 10,000, 100,000, etc. Because 100 = 10², 1000 = 10³, etc., the above series of numbers can be written as: 10¹, 10², 10³, 10⁴, 10⁵, etc. The superscripts 1, 2, 3, 4, 5, etc. are called powers, or exponents, of the base 10. If we always use 10 as the base, the exponents are called logs to the base 10. By pairing the original numbers with the exponents, we have the beginning of a table of logs (see Table C-1).

This table can be extended indefinitely downward. Since one million contains six tens as factors (10 X 10 10 X 10 X 10 X 10), the log of 1,000,000 is 6. For any number containing only the digit 1 (other than zero), the log can be found by counting the number of decimal places between the position to the right of the 1 and the position of the decimal point. The log of 1,000,000,000 is therefore 9. The number whose log is 5 is 100,000, and 100,000 is said to be the antilog of 5. If the columns in Table C-1 are extended upward one step, the number 1 is added to the first column and the log 0 is added to the second. Thus the log of 1 is 0 and the antilog of 0 is 1.

Table C-1

Number	Logarithm
10	1
100	2
1,000	3
10,000	4

We also need the logs of numbers between 1 and 10. Since the log of 1 is 0 and the log of 10 is 1, the logs of numbers between 1 and 10 must be decimal fractions between 0 and 1. The logs of numbers from 1 to 10 are listed in Table C-2. It is customary to write the log of a number to one more significant figure than the number itself. Thus the number 8 has one significant figure, and 0.90 (the log of 8) has two significant figures.

Table C-2 is used to illustrate some basic relationships between numbers and their logs:

1. 6 = 2 × 3. Note that the log of 6 is the sum of the logs of 2 and 3. When numbers are multiplied, their logs are added. The log of 10 equals the sum of the logs of 2 and 5.
2. 4 = 8 ÷ 2. Note that the log of 4 is the difference of the logs of 8 and 2. When one number is divided by another, the log of the second is subtracted from that of the first. The log of 3 equals the difference of the logs of 6 and 2.

   Table C-2

   Number	Logarithm
   1	0.00
   2	0.30
   3	0.48
   4	0.60
   5	0.70
   6	0.78
   7	0.85
   8	0.90
   9	0.95
   10	1.00

3. 4 = 2². Note that the log of 4 is twice the log of 2. When a number is squared, the log is doubled. Similarly, 8 = 2³, and the log of 8 is three times the log of 2. When a number is raised to a power, the log of that number is multiplied by the power.
4. √9 = 3. Note that the log of 3 is half the log of 9. Similarly, 2 is the cube root of 8, and the log of 2 is one-third the log of 8. When a root is taken of a number, the log of the number is divided by the root.

By the multiplication rule, the log of 50 is the log of 5 + the log of 10, or 0.70 + 1.00, or 1.70. Similarly, the log of 500 is 0.70 + 2.00, or 2.70; the log of 5000 is 3.70, etc. Also the log of 200 is 2.30, the log of 300 is 2.48, etc. All numbers in the hundreds have logs beginning with 2, all numbers in the thousands have logs beginning with 3, etc.

From this comes the concept that the log of a number consists of two parts: a whole number, determined by whether the original number is in the tens, hundreds, thousands, etc.; a decimal part determined by the digits. The whole number part of a logarithm is called the characteristic; it is solely determined by the location of the decimal point in the original number. The decimal part of a log is called the mantissa; it is found from a table of logs. To find the characteristic, count the number of places from a spot just to the right of the first digit of the number to the decimal point. For 500,000, the count is 5, and therefore the log begins with 5. The decimal part of the log, found in Table C-2, is 0.70, and the entire log is thus 5.70.

If a log is given, the antilog (the number corresponding to the log) is found by the reverse process—that is, use of the decimal part of the log to find the digits in the number and use the characteristic of the log to place the decimal point in the number. For example, to find the antilog of 3.78, note that 0.78 is the log of the number 6 in Table C-2. Therefore, 6 is the antilog of 0.78. The 3 in the log 3.78 indicates that the decimal point is moved three places to the right, changing the number from 6 to 6000.

We will now extend the basic log table to include decimal numbers between 0 and 1. In Table C-3, each number has one-tenth the value of the number just below it, whereas each log is reduced by a value of 1 for each step upward. Thus, the next number above 1 will be one-tenth of 1, or 0.1, and the log will be 1 less than 0, or -1, and so on. The basic log table can now be considered to extend indefinitely in both directions. Note the symmetry in Table C-3 about the number 1: for example, all numbers in the thousands have logs containing 3, and all numbers in the thousandths have logs containing -3.

Table C-3

Number	Logarithm
0.0001	-4
0.001	-3
0.01	-2
0.1	-1
1.0	0
10.0	1
100.0	2
1000.0	3
10000.0	4

(Note that the number column is identified as a ratio scale, which will never reach zero, and that the log column is identified as an interval scale.)

The logs of decimal fractions are found by the same procedure as that used for larger numbers. To find the characteristic (the whole number part of the log), count the number of places from the position to the right of the first nonzero digit to the decimal point. To find the mantissa (the decimal part of the log), locate the nonzero digit in the number column and note the corresponding value in the log column. For example, to find the log of 0.0003, the decimal point is 4 places to the left of the digit 3, so the log has a characteristic of -4. The mantissa is 0.48, opposite 3 in the number column in Table C-2.

The most awkward thing about logs is that there are several ways of writing the logs of numbers between 0 and 1. Most often the mantissa (a positive value) and the characteristic (a negative value) are kept separate and are written as an indicated unfinished computation. For example, three ways of writing the log of the number 0.004 are:

1. 0.60 -3, where 0.60 is the mantissa and -3 is the characteristic. This is the most direct form, but it is not often used.
2. 7.60 -10, where 0.60 is the mantissa and the combination 7 - 10 (which is equal to -3) is the characteristic. This form, which is always written as some number minus 10, is the form most commonly used outside photography.
3. 3.60, where 0.60 is the mantissa and 3 is the characteristic. The minus sign is placed over the 3 to emphasize that only the 3 is negative, not the 0.60. This form is referred to as bar notation, and the log above is read “bar three point six zero.” This form is found in many photographic publications, but the scientific calculator is making bar notation obsolete.
4. With a scientific the log of 0.004 can be written as -2.40.

The procedure for finding the antilog of a log having a negative characteristic is the same as described above for positive logs, except that the decimal will be moved to the left in the number rather than to the right. With 3.60 as the log, for example, the antilog of 0.60 is 4, and moving the decimal three places to the left produces 0.004. Thus, the antilog of 3.60 is 0.004.

An expanded table of logs for numbers from 1.0 to 10.0 to one decimal place, with logs to three decimal places, is provided in Table C-4.

Logarithms and the Use of a Calculator

The use of scientific hand calculators greatly eases the tasks of determining logs and anti- (or inverse) logs. To find the log of a number greater than 1, simply enter the number and then press the key labeled “LOG.” The answer will appear on the display. To

Table C-4 Abbreviated table of logarithms

Number	Logarithm	Number	Logarithm	Number	Logarithm
1.0	0.000	5.5	0.740	0.0001	-4
1.1	0.042	5.6	0.748	0.001	-3
1.2	0.080	5.7	0.756	0.01	-2
		5.8	0.763	0.1	-1
1.26	0.100	5.9	0.771	1	0
				10	1
1.3	0.114	6.0	0.778	100	2
1.4	0.147	6.1	0.785	1000	3
		6.2	0.792	10,000	4
1.414	0.150	6.3	0.799	100,000	5
		6.4	0.806
1.5	0.175			2	0.301
		6.5	0.813	20	1.301
1.6	0.204	6.6	0.819	200	2.301
1.7	0.230	6.7	0.826	2000	3.301
1.8	0.255	6.8	0.832
1.9	0.278	6.9	0.839
2.0	0.301	7.0	0.845		0.301-1
2.1	0.322	7.1	0.851		or
2.2	0.342	7.2	0.857	0.2	9.301-10
2.3	0.361	7.3	0.863		or
2.4	0.390	7.4	0.869		1.301
					or
2.5	0.398	75	0.875		-0.69
2.6	0.415	76	0.881
2.7	0.431	77	0.886
2.8	0.447	78	0.892
2.9	0.462	79	0.898
3.0	0.477	8.0	0.903
3.1	0.491	8.1	0.908
3.2	0.505	8.2	0.914
3.3	0.518	8.3	0.919
3.4	0.532	8.4	0.924
3.5	0.544	8.5	0.929
3.6	0.556	8.6	0.934
3.7	0.568	8.7	0.940
3.8	0.580	8.8	0.944
3.9	0.591	8.9	0.949
4.0	0.602	9.0	0.954
4.1	0.613	9.1	0.959
4.2	0.623	9.2	0.964
4.3	0.634	9.3	0.968
4.4	0.644	9.4
4.5	0.653	9.5	0.978
4.6	0.663	9.6	0.982
4.7	0.672	9.7	0.987
4.8	0.681	9.8	0.991
4.9	0.690	9.9	0.996
5.0	0.699	10.0	1.00
5.1	0.708
5.2	0.716
5.3	0.724
5.4	0.732

Table C-5

Number	Bar Log	Negative Log
0.0002	4.3	-3.7
0.002	3.3	-2.7
0.02	2.3	-1.7
0.2	1.3	-0.7
2	0.3	0.3
20	1.3	1.3
200	2.3	2.3
2000	3.3	3.3

find the antilog of a positive log, simply enter the log value and press the keys labeled “INV” and “LOG” in that order. The answer will appear on the display.

Note: On some calculators, there is no key labeled “INV.” For these calculators, the key labeled “10^x” should be substituted for the entire “INV” and “LOG” key sequence.

Example

Problem: Find the log of 0.004.
Solution: Enter the number 0.004 into the calculator and press the key labeled “LOG”; the answer of -2.39794 rounded to -2.40 is shown in the display.
Answer: The log of 0.004 is -2.40.

To find the antilog of a totally negative log, the above procedure is reversed. The totally negative log is entered into the calculator and the keys labeled “INV” and “LOG” are pushed, in that order.

Example

Problem: Find the antilog of -2.40.
Solution: Enter the value -2.40 into the calculator and press the keys labeled “INV” and “LOG” in that order; the number 0.003981 appears in the display and is rounded to 0.004.
Answer: The antilog of -2.40 is 0.004.

Since the bar notation system is often encountered in photographic publications, a procedure for using a calculator with this system is given below.

To find the log of a number less than 1.0 expressed in bar notation using a calculator:

1. Move the decimal point to the right of the first nonzero digit and count the number of places it was moved. The number of spaces moved is the characteristic it will be, but assigned a negative value.
2. Enter the number (with the decimal moved) in the calculator and press the button labeled “LOG”; the value that appears in the display is the mantissa of the number and will be a positive value.

Example

Problem: Find the log of 0.2.
Solution:

1. The decimal point is moved one place to locate it to the right of the first nonzero digit (2); thus the characteristic is bar one (1).
2. Enter the number 2.0 in the calculator and press the button labeled “LOG”; the value of 0.30 appears and is the mantissa.

Answer: The log of 0.2 is 1.30.

To find the antilog (inverse log) of a log expressed in bar notation using a calculator:

1. Enter the mantissa of the log into the calculator and press the buttons labeled “INV” and “LOG,” in that order; the value that appears on the display will contain all of the digits of the antilog with the decimal point immediately to the right of the first digit.
2. Locate the decimal point at the correct position by moving it to the left the number of places indicated in the characteristic.

Example

Problem: Find the antilog of 1.30.
Solution:

1. The mantissa of 0.3 is entered in the calculator and the keys labeled “INV" and “LOG" are pushed and the value 1.9952623 appears, which is rounded to 2.0
2. The characteristic is bar one, indicating that the decimal should be moved to the left one position, giving 0.2.