If the number consist of less than five figures, we proceed by RULE III. With the given number enter the column marked N; and opposite will be found the mantissa. To this prefix the characteristic. The result will be the required logarithm. If the number consist of more than four figures, RULE IV. Find the mantissa corresponding to the first four figures. With the additional figures, one by one, enter the column PP under the proper tabular difference, and add the corresponding proportional parts to the mantissa first found. Prefix the characteristic. The result will be the required logarithm. EXAMPLES. 1. Find the logarithm of 24536. Mantissa of 2453 = 38970 = 17 log 24536 4.38980% 2. Find the logarithm of 189637. Mantissa of 1896 = 27784 23 69 161 16.1 If the logarithm be given, the number which corresponds to it is found by the following rules. When the mantissa can be found in the tables, we proceed by RULE V. 1o. With the given mantissa enter the column marked Log. 2°. Take from the column N the corresponding four figures. 3°. From the characteristic, find the position of the decimal point. The result will be the required number. EXAMPLES. 1. Given the logarithm 2.93120; find the number. 2. Given the logarithm 1.72534; find the number. 3. Given the logarithm 5.54986; find the number. Ans. 853.5. Ans. 53.13. Ans. 354700. 4. Given the logarithm 2.85986; find the number. Ans. 0.07242. 5. Given the logarithm o.54741; find the number. Ans. 3.527. When the mantissa cannot be found in the tables, we proceed by RULE VI 1o. Find the next lower mantissa, and note the four corresponding figures in column N. 2. With the difference between this and the proposed mantissa, enter the table of proportional parts, and find the additional figures. 3°. Place the decimal point according to the characteristic. EXAMPLES. 1. Given the logarithm 2.31989; find the number. With the difference 16 we enter the column PP under the tabular difference 21, and find that 7 is the first additional figure, as it corresponds to 14.7, the proportional part next lower than 16; the difference between this number and 16, when multiplied by 10, i. e. 13, shows that 6 is the next figure, as it corresponds to 12.6, which is next lower than 13. To the figures 2088 must therefore be annexed 76, and as the characteristic is 2, the number required is Ans. 208.876. 2. Given the logarithm o.19237; find the number. Ans. 1.55728. 3. Given the logarithm 1.56342; find the number. Ans. 0.36595. 5. Multiplication by Logarithms.—In multiplication we proceed by RULE VII. 1°. Find the logarithms of the numbers, the product of which is required. 2°. Add these together; the sum will be the logarithm of the product. 3°. Find from the tables the corresponding number. This will be the required product. 6. Division by Logarithms.—In division we proceed by the following RULE VIII. 1°. Find the logarithms of the numbers the quotient of which is required. 2°. Subtract the logarithm of the divisor from that of the dividend; the difference will be the logarithm of the quotient. 3°. Find from the tables the corresponding number. This will be the required quotient. 7. Involution by Logarithms.—In involution we proceed by the following RULE IX. 1o. Find the logarithm of the number which is to be raised to the given power. 2o. Multiply this logarithm by the index of the given power; the product will be the logarithm of the required power. 3°. Find from the tables the corresponding number. This will be the required power. |