Elements of analysis as applied to the mechanics of engineering and machinery . ing to the base a. Therefore, by help of the same, thenatural logarithm may be changed into any artificial logarithm, andvice versa. For Briggs^ system of logarithms, the base is a = 10,hence -^ = nat. log. 10 =: 2,30258 . ., and inversely, the modulus: m = ——- = 0,43429 . ., nat. log. 10 therefore, log. y r= 0,43429 nat. log. ?/, andno,t. log. y = 2,30258 log. 20. The course of the curves whicli correspond to the expo-nential functions y =^ e and ?/ = 10, is illustrated in Fig. 32. Art. 20.] ELEMENTS OF AT^
Elements of analysis as applied to the mechanics of engineering and machinery . ing to the base a. Therefore, by help of the same, thenatural logarithm may be changed into any artificial logarithm, andvice versa. For Briggs^ system of logarithms, the base is a = 10,hence -^ = nat. log. 10 =: 2,30258 . ., and inversely, the modulus: m = ——- = 0,43429 . ., nat. log. 10 therefore, log. y r= 0,43429 nat. log. ?/, andno,t. log. y = 2,30258 log. 20. The course of the curves whicli correspond to the expo-nential functions y =^ e and ?/ = 10, is illustrated in Fig. 32. Art. 20.] ELEMENTS OF AT^ALYSIS. 31 For ^ == 0, Tre have in both cases y^=e^^=a^=l\ hence, also, bothcnrves OQS and OQ^S^ pass throngh the same point {0) m theaxis of orclinates AY. For ^ := 1, we have y = e^ = 2,tl8 . . and 2/ = 10^ = 10,for X = 2, y = e^ = 2,n8^ = T,389 and y = 10^ = 10^ = 100, &c.;therefore, on the positive side of the axis of abscissas, both cnrvesascend very perpendicnlarly; and particularly the last. On the otherhand we have, for x =^ — 1: * - = . . and. ^ 2,n8 . = 10-1 = 0,1; 2; 0,135 and 10^ further, for x = • pX p—1 ^ -^ ~2,n8^ = 10-2 = 0,01; and, lastl}^, forX =^ ?— 00, both equations give; e- = — = — = 0. Consequently, on the negative sideof the axis of abscissas, the twocurves approach nearer and nearerto the same, the latter, indeed, moreabruptly than the former, but anactual coincidence with the axisnever takes place. As from y = e^, there resultsX. = nat. log. ?/, and likewise from y X = log. ?/, these curves furnish a scale of the natural loga-rithms, and of those of Briggs ; theabscissas are, ^dz., the logarithmsof the ordinates, and we have, forexample, AM= nat. log. MP= log.^ MP^, & to the differential for- 2 —1 T AM 1 2^—^ mula lY. of the last article, the tangential angle a of the exponential curve is determined by thesimple formula: 6?y a^ dx a^ y tang, a = ^^ = -— cx m cx 111 on = y nat. log. a. 32 ELEMENTS OE ANALYSIS. [Art
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