What is here said upon the subject of logarithms is doubtless sufficient to convey to the student a correct notion of their nature and properties, as also of the practicability of constructing a table of them to any extent. The labour, however, of actually computing a whole table of logarithms by means of the series here investigated, would be great in the extreme; they are, however, susceptible of a variety of transformations much better adapted to the use of the computer. To explain and exhibit these would carry us too far into the business of series, and would occupy too large a portion of this treatise. But the inquiring student, who is desirous of ample information upon the most expeditious methods of calculating a table of logarithms, may refer to the second edition of the author's " Essay on the Computation of Logarithms ;" and the manner of using a table thus constructed is fully explained, in the introduction prefixed to the "Mathematical Tables." APPLICATION OF LOGARITHMS. LOGARITHMICAL ARITHMETIC. (151.) From what has been already said on the nature and properties of logarithms, the following operations, performed by means of a table, will be readily understood without any further explanation. EXAMPLE 1. Multiply 23.14 by 5.062. Here the log. of 23·14 in the tables* is 1.3643634 66 * The tables employed are Young's " Mathematical Tables,” computed to seven places of decimals. The few examples here given are sufficient to show the great advantage of logarithms in abridging arithmetical labour, in which indeed consists their principal, although not their only value. There are many analytical researches which it would be impossible to carry on without their aid, and many others in which the introduction of logarithmic formulas greatly facilitates the deductive process. It would be easy to propose questions, the solutions of which might be comprised in a few lines by logarithms, but which, without their aid, would occupy many volumes of closely printed pages. The following is a striking example. Let there be a series of numbers commencing with 2, and such that each is the square of that which immediately precedes it: it is required to determine the number of figures which the 25th term would consist of. The series proposed is obviously 2, 22, 24, 25, 26, &c. the exponent of the nth term being 2"-1, and consequently the exponent of the 25th term is 224 16777216; consequently, calling the 25th term x, we have = x=216777216, whence log. x = 16777216 log. 2 = 16777216 × ⚫30103 hence, since the index or characteristic of this logarithm is 5050445, the number answering to it must consist of 5050446 figures, so that the number x, if printed, would fill nine volumes of 350 pages each, allowing 40 lines to a page, and 40 figures to a line. ON EXPONENTIAL EQUATIONS. (152.) An exponential equation is an equation in which the unknown term is expressed in the form of a power with an unknown index; thus, the following are exponential equations: a* = b, xx = a, ab*= c, &c. (153.) When the exponential is of the form a*, the value of x is readily found by logarithms; for of ab, we have Also, if ab= c, put b2 = y, then a3 = c, whence y log. a = = log. c; (154.) But if the equation be of the form a = a, then the value of a may be obtained by the following rule of double position. Find by trial two numbers as near the true value of x as possible, and substitute them separately for x, then, as the difference of the results is to the difference of the two assumed numbers, so is the difference of the true result, and either of the former, to the difference of the true number and the supposed one belonging to the result last used; this difference therefore being added to the supposed number, or subtracted from it, according as it is too little or too great, will give the true value nearly. And if this near value be substituted for x, as also the nearest of the first assumed numbers, unless a number still nearer be found, and the above operations be repeated, we shall obtain a still nearer value of x, and in this way we may continually approximate to the true value of x. 1. Given x= and EXAMPLES. 100, to find an approximate value of x. upon trial x is found to lie between 3 and 4; .. substituting each of these, we have 3 log. 3 1.4313639 = Now this value is found, upon trial, to be rather too small; and 3.6 is found to be rather too great; therefore, substituting each of these, we have 3.582 log. 3.582 = 1.9848779 3.6 log. 3.6 =2.0026890 0178111 diff. of results. .. 0178111: 018 :: 002689 : 002717, whence 3.60027173.597283 = x very nearly. The operation of solving the equation * a may be conducted differently, by using logarithms throughout; thus, in the equation x log. x log. a, call log. x, x'; and log. a, a'; then xx'a',.. = log. x + log. x' = : log. a', that is, x' + log. x' = = log. a'; hence we have to find a number x', which, when increased by its log., shall be equal to log. a', which may be effected by the rule of position before given. Thus, taking the same example as before, viz. r* = 100, we have log. 100=2=a', and log. 2=3010300; .. x'+log. x'=·3010300, and the nearest value of x' in the tables below the true value is ⚫55597, which added to its log. 1-7450514, gives 3010214, and ... the nearest value above the truth is 55598, which added to its log. 1·740592, gives 3010392; hence, by the rule: 3010392 3010214 178 3010300 3010214 86 .. 178 : 1 :: 86: 483, consequently '55597483 = log. x, . x 3.597284. 1 If a be less than unity this solution fails, since a' is then negative, and therefore the log. a' is unassignable. But if we put x = and " y we shall have, by substitution, the equation by y,.. y log. α= 1 b' b= log. y; put log. b = b', and log. y = y', then yb' = y', .. log. y + log. b' log. y', or y' + log. b' = log. y'; whence y' may be found by the preceding rule. = 2. Given a = 5, to find an approximate value of x. Ans. x 2.1289. 3. Given 2000, to find an approximate value of x. COMPOUND INTEREST. (155.) INTEREST is a certain sum paid for the use of money for any stated period, and when the interest of this money is regularly received, the money, or principal, is said to be at simple interest; |