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 help you to save time in multiplication, division, raising a number 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.