The Logarithm as a Direct Function |
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analysis appear approaches zero area Arithmetik Assuming Ax approaches ax+y base becomes negatively infinite becomes positively commensurable continuous derivative continuous function curve deduce defined Definition difference quotient Differential DIRECT FUNCTION elementary calculus equation exists Exponential Function fixed following theorems formula FUNCTION By J. W. function F(x function of x functional relation Fundamental Property given hence increases indefinitely integral interval Inverse J. W. BRADSHAW January limit log.b loga lying mathematics means once point positive integer positive number positive qth root positive value power proof Property of Logarithms rational number rational values relation xy rigorous SALEM PRESS satisfies second function shown shows Similarly single valued function solution Stolz students Substituting sufficient Suppose tive treatises valued and continuous values of x variable ordinate wish to prove x approaches zero x-axis xy dt
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Page 53 - The logarithm of a number is the exponent of the power to which it is necessary to raise a fixed number, in order to produce the first number.
Page 53 - The logarithm of any number to a given base is the index of the power to which the base must be raised in order to equal the given number.
Page 55 - Therefore ф(х) becomes positively infinite as x increases indefinitely, and thus the first part of the theorem is proved. To prove the second part of the theorem, put ¿
Page 53 - ... of students of the Calculus never see a proof that there is such a thing as a logarithm. It is possible, however, to supply a proof by means of elementary calculus, inclusive of the general theorems about continuous functions with which all students are familiar. How simple the analysis is appears from a casual glance at the following pages, in which Mr. Bradshaw has carried through all the details of a rigorous development of the essential properties of the Logarithm and, as it appears here,...
Page 53 - It is hoped that this presentation may prove attractive to students who have finished a thorough course in elementary calculus. w. F. o. * Stolz, Allgemeine Arithmetik, vol.
Page 57 - ... analytic proof may be given by means of the following well known theorem of continuous functions. As x varies continuously from a to b, any function f(x), continuous in the interval...
Page 60 - ... (4) * This method has been employed in the study of the logarithmic function for complex values of the argument by Burkhardt, Analytische Funktionen, p. 162. ф(х) + ф(х) + • • • to н terms = ф(х • х • • • • to n factors), .-. пф(х)=ф(х*), (5) ?! being a positive integer. Assuming, as we have in the last equation, the definition of a" when n is a positive integer, we can prove the THEOREM.
Page 62 - Substituting e for n in (14) we have a second functional relation )]». (16) 7. The Sufficiency of the Functional Relations. It turns out that the functional relations...