8. Using r=3.14, find the radius (r) of a sphere whose volume (V) is 1,975 cubic inches. The formula for finding the volume 4 a ХтXr of a sphere is V=1X*XX. 9. Using the formula for the volume of a sphere given in problem 8 above, find the volume in cubic feet of a sphere having a radius of 21 inches. а 10. A rectangular swimming pool is 135 feet long and 78 feet wide: How much further is it to swim around the edge from one corner to the diagonally opposite corner than it is to swim directly between the same two corners? The end-of-course test, for which you recently applied, may either be taken at this time or after lesson 12, which is optional. Lesson 12 Logarithms This lesson will be devoted to explaining logarithms and their use in laying out the scales of a slide rule. The text assignment picks up the sections on logarithms which were previously omitted. Although you are not required to study this lesson, it is suggested you at least read it over once since it contains much interesting and valuable material. Text Assignment 38-42, 60-65, 82-84, 118-121, 151-155, 160-161, 172, a Study Notes To understand logarithms it is necessary to be familiar with the laws of exponents. As a matter of fact, a particular use of exponents is so important that exponents used in this way are called logarithms. The logarithm of a positive number to a given base, other than 1, is the exponent to which the base must be raised to equal the number. Thus, if a positive number N is expressed as a power of b by means of the equation N=6", the exponent x is called the logarithm of the number N to the base b. This can be expressed as x=log,N. Therefore, a logarithm is an exponent and exponential equation N=b* means the very same thing as logarithmic equation x=logoN. To illustrate, 23=8 and log2 8=3 mean the same thing. Since logarithms are exponents, the laws which govern operations with logarithms are the same as those which hold true for exponents. Certain laws of exponents are shown below with a specific example for each. Law for multiplication 64.bv=6*+y Example, 23.22=23+2= 25 Law for a power of a power (64)=bav Example, (23)2=23.2=28 The properties of logarithms which are comparable to the above laws of exponents are given below. The logarithm of a product is equal to the sum of the logarithms of its factors, all logarithms being taken to the same base. logoMN=logoM+logoN The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor, all logarithms being taken to the same base. м 3 logon a The logarithm of a power of a number is equal to the exponent times the logarithm of the number, all logarithms being taken to the same base. logoMP=p logoM As you can see from the general properties above, any base can be used. However, 10 is the most convenient base for computational purposes. It is only when 10 is used as the base that the mantissa of the logarithm depends solely on the sequence of the digits of the number. When 10 is used as the base, it is referred to as the common system of logarithms. Later in this lesson the system of natural logarithms is discussed. This system uses e, which has a value of 2.71828 to 5 decimal places, as its base. Thus, loge 7 means the logarithm of 7 to the base e. Similarly, log, 7 means the logarithm of 7 to the base 4. On the other hand, log 7 means the logarithm of 7 to the base 10. In other words, when the base is not expressed, it is understood to be 10. Pages 60–64. The first property of logarithms given above states that the logarithm of the product of two numbers is equal to the sum of the logarithms of the two numbers. If you read these pages care fully, you will see that when you multiply two numbers using the C and D scales you are actually adding their logarithms to get the product. Pages 82–84. On these pages your text explains how the slide rule finds the quotient of two numbers by subtracting the logarithm of the divisor from the logarithm of the dividend. This is in keeping with the second property of logarithms given above. Pages 118–121, 151–155. On these pages your text discusses how the slide rule employs logarithms to find the square, square root, cube, and cube root of a number. The third property of logarithms given above is used for these operations. For example, in logarithmic 2 form, 32, 17, 5', and ¥12 can be expressed as 2 log 3, į log 7, 4 log 5, and log 12, respectively. 1 3 Page 193. In both using the slide rule and in understanding its operation it was never actually necessary to find the logarithm of a number. Many calculations, particularly those in engineering, require the use of logarithms. Logarithms to the base 10 can be obtained from the log scale of the slide rule. Page 196. Pay particular attention to the second paragraph which explains the method of writing logarithms for numbers less than 1. It has been stated previously that the mantissa of a logarithm to the base 10 depends only upon the sequence of the digits. This is true because the mantissa is always written as zero or as a positive number less than 1. Page 198. Be sure you understand what is meant by antilog. When you are asked to find the antilog, you are given the logarithm of a number and must find the number. For example, if you are asked to find the antilog of 5.4771, you must find the number which has 5.4771 as its logarithm. To do this, turn the slide over and set 0.4771 on the log scale under the hairline of the celluloid insert. Next, turn the slide over again and read the digit or digits on the C scale over the right index of the D scale. In this case there is one digit, 3. Since the characteristic of 5.4771 is 5, the digit count is 6. The answer is 300,000. Page 202. To find the logarithm of a number to a base other than 10 by means of a slide rule, you must first find the logarithm of the number to the base 10 and then multiply it by a constant. This constant is determined by the base being used. When e is the base the constant is 2.3026. Thus, to find log. 91, first find log 91 which is 1.959. Next, multiply 1.959 by 2.3026 to get 4.51, which is the logarithm of 91 to the base e. To find the antilog of a logarithm to the base e, you must divide the logarithm to the base e by 2.3026. This will give you the logarithm of the number to the base 10, which can be found by the method described earlier in this lesson. To illustrate, suppose you wish to find the number which has 1.275 for its logarithm to the base e. Dividing 1.275 by 2.3026 gives 0.554, which is the logarithm of the number to the base 10. By setting 0.554 on the log scale, the number can be found on the C scale over the right index of the D scale. It is 3.58. Page 203. The answer given in the text for Practice Problem 4 is more exact than you probably will get using the slide rule. Your answer will probably be – 0.476. Page 205. Your answer to Practice Problem 7 will probably be 19.1 instead of 19.13 as given in the text. Page 206. Check your answers to the Review Problems with those given at the back of this study guide. Pages 211-213, 217-220. The material on these pages explains how the CI scale in conjunction with the D scale actually adds logarithms to multiply and subtracts logarithms to divide. This is similar to the explanations given earlier regarding multiplication and division with the C and D scales. Self-Examination Select the correct answer in each of the following exercises. d. 2.776. b. 1.776. e. none of the above. c. 2.396. 2. The logarithm of 4,043 to the base 10 is a. 3.254. d. 4.607. b. 3.607. e. none of the above. c. 3.646. 3. The logarithm of 0.000295 to the base 10 is a. 6.470—10. d. -3.470. b. 7.192-10. e. none of the above. c. 7.470–10. |