2IO Algebra. 



J. Find the tenth power of .69353, whose logarithm is 

 1. 8410653. 



/. Find the seventh power of 15.926, whose logarithm is 

 1. 2021067. 



27. Examples IN Evolution. See Art. 10. 

 /. Find the cube root of 26. 



From a table of logarithms we find 



log 26 = 1.4141374 

 Therefore log ^26=0.4713791 



The number whose logarithm is 0.471 3791 is found to be 

 2.9606 + 



Hence ^^26=2.9606-1- 



2. Find the square root of 668.63, whose logarithm is 

 2.8251859. 



J. Find the fifth root of 11 09600000000, whose logarithm 

 is 12.0451664. 



^. Find the tenth root of 1.384, whose logarithm is 

 0.1411675. 



28. Exponential Series. The Exponential Series, or the 

 Exponential Theorem, as it is often called, is an expansion oi a"- 

 in terms of the ascending powders of x. The following demonstra- 

 tion* of this important theorem is due to Mr. J. M. Schaeberle, of 

 the Lick Observatory, and is inserted here w4th his permission. 



We are required to expand a"" in a series of ascending powers of 

 X. Assume 



^^ = A-hB.t-+Ct:^ + Djt-3-^E-:i-'+ ... (i) 



where A, B, C, etc., are undetermined coefficients. 



The limit of the left-hand side of this equation as x approaches 

 o is plainly i. The limit of the right-hand side of this equation 

 as X approaches o is A. (See XI, Art. 26.) 



Therefore A=i. 



Substituting this value of A in (\) and then, squaring both 

 members, 

 ^-=I + 2BJl.--h(2C-fB•0-^"+(2D-f 2CB).r^ 



-h(2E-h2DB + 0^-^ . . . (2) 



*See Annnls of Mathematics, Vol. Ill, p. 15J. 



