Here because the 9 lies between 1-A, and 10B, find a mean proportional C between them, and the logarithm of the fame, is half the fum of the two laft logarithms. In like manner, is found, a mean proportional D between B and C; likewife the logarithm of D is half the sum of the logarithms of B and C; fo in the 18th step of this procefs, the logarithm of 9, is found to be 0,954242. When the logarithms of prime numbers are thus calculated, the bufinefs becomes easier; for the logarithms of compofite numbers may be obtained, by adding the logarithms of their component parts. Thus, the logarithm of 15 may be found, by adding the logarithm of 3 and 5 together; for 3+5=15, and so on of any other compofite number. The logarithms of roots are raised to any given power, by multiplying them by the exponent of the power, & vice versa. PROBLEM I. To find the logarithm of any given number from the tables. IT is ufual to divide logarithmic tables into 10 columns: In the left hand column, are the natural numbers between 100 and 1000, and at the top and bottom are marked, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. If the natural number is less than 100, its logarithm is found in the first page. If it exceed 100, and is lefs than 1000, the number is found in the left hand column, marked No. and its logarithm is found oppofite to it in the adjacent column, under o; but if the number exceed 1000, and is lefs than 10000, find the three highest figures in the column of numbers, and in the fame line, titled by the unit at the top, is the logarithm required. The logarithm of 1786, may be found from the tables: thus, in the column No. look for 178, and in the fame line, under 6, (the units place at he top) is 3.25188, the logarithm required. Note, In every cafe, the index is lefs by 1, than the number of places; and, on the contrary, the number of places i's greater by unity than the index. The logarithms of mixed numbers, are found the fame as if they were integers; but the integer alone determines the in dex. Decimal fractions have negative indices, which are to be added when the logarithms are fubtracted, and subtracted when the logarithms are added. PROBLEM II. To find the natural number correfponding to a given logarithm. LET the given logarithm be 2.75976, it is required to find its correfponding number. Look for the given logarithm, neglecting the index, and against it on the margin, you find 575, and 1 at top, which is 5751; but the index being 2, the integer must therefore confift only of three places; and, by pointing off towards the right hand for decimais, the number will be 575.1. It often happens, that the exact logarithm cannot be found in the tables, in which cafe we take the nearest to it. PROBLEM III. To find the product of two given numbers by logarithms. Rule, Add the logarithms of both factors together, and their fum is the logarithm of the product. Ex. Res Ex. Required the product of 15, multiplied by The logarithm of 15, is 1.17609. The log. of 1050, the product, 3.02119. 1 PROBLEM. IV. To find the quotient of two given numbers by logarithms. Rule, From the logarithim of the dividend, fubtract the logarithm of the divifor, and the remainder is the logarithm of the quotient. Ex. Required the quotient of 425, divided by 15. The log. of 425, is 2.62839. The log. of 15, is 1.17609. The log. of 28.33, the quotient 1.45230. PROBLEM V. To find the fquare, cube, or any higher power of a given number, by logarithms. Rule, Multiply the logarithm of the root, by the exponent of the power, and the product is the logarithm of the power required. Ex. Required the cube of 12. The log. of 12, is 1.07918. 3. The log. of 123=1728-3.23754. PRO PROBLEM VÍ. To extract the fquare, cube, biquadrate, &c. root of a given number by logarithms: Rule, Divide the logarithm of the given number, by the exponent of the power, and the quotient will give the logarithm of the root. Ex. Required the cube root of 1728. The logarithm of 1728, is 3.23754, which, if divided by 3, will quot 1.07918, the logarithm of 12 the root PROBLEM VII. Three numbers being given, to find a fourth proportional to them: Rule, From the fum of the logarithms of the fecond and third terms, fubtract the logarithm of the first, and the remainder is the logarithm of the answer.. Ex. If 14 yards cloth, coft 7 I., what will 70 yards cost at that rate? The log. of 14, is 1.14613 first term. of 7, is 0.84510 fecond term. of 70.5, is 1.84819 third term. 2.69329, fum of the 2d and 3d terms. Log. of 35.25, is 1.54716. remainder. or 35 L. 5s. PRO PROBLEM VIII. To find a mean proportional between any two numbers by logarithms. Rule, Add the logarithms of the two given numbers together, and half their fum is the logarithm of the mean proportional. Ex. Required a mean proportional between 8 and 32. The log. of 8, is 0.90309. The log. of 32, is 1.50515. for 8: 16: 16:32. 2) 2.40824. The log. of 16, the mean prop. 1.20412. L PROBLEM IX. To find the logarithm of the Sine, Tangent, Secant, belonging to any number of degrees and minutes required. Rule, If the degrees required, be less than 45°, seek the degrees on the top, and the minutes in the left hand column titled M, in the fame line under the propofed name at the top, ftands the fine, tangent and fecant required. If the degrees given, exceed 45°, feek the degrees at the bottom, and the minutes in the right hand column marked M, and the proposed name at the bottom. Note, If the degrees at the top and the minutes in the left hand column, be added to the degrees at the bottom and minutes in the right hand column, the fum will be 90°. Hence they are complements of each other. |