The Natural LogarithmFor real numbers, the following sections describe the area-under-a-curve definition of the natural logarithm, and how this introduction of the natural logarithm leads to the definition and properties of all logarithms, exponentials and powers involving real numbers.The natural logarithm ln(a) for a > 0 can be introduced FOOTNOTE: Variants of the exposition given here may be presented less cryptically in other texts. The presentation here is show briefly the approach I would like to see favored in schools. Working through the details of this exposition in its present form could be a subject for discussion in a high school math club. Understanding this section and the next demands or provides a sound command of some mathematics beyond arithmetic.as the (signed) area under the curve y = [1/(s)] from s = 1 to s = a. Equivalently, it may be represented by the signed area under the curve u = [1/(v)] from v = 1 to v = a. This definition does not depend on the labelling of the horizontal and vertical axes. See the next two diagrams. In the next diagram, the area from s = 1 to s = a > 1 can be approximated by slicing it into n vertical rectangles with the same base size [(a-1)/(n)], and then making this base size smaller by letting n®¥ (that is get larger and larger). The sum of the area of the resulting rectangles approximates to a single number L with greater and greater accuracy, more decimal places say, as n ®¥. This single limit gives what we call ln(a).
For a ³ 1, the value of ln(a) is given by the area from s = 1 to s = a under the curve y = [1/(s)]. Here we take or assume ln(1) = 0. It can be shown that ln(a) ® 0 when when a approaches 1 through values above or greater than 1. The natural logarithm ln(b) of a number b when 0 < b < 1 is defined next.
For 0 < b < 1, the value of ln(b) is given by (-1) times the area under the curve y = [1/(s)] from s = b to s = 1. The above two diagram illustrate the arithmetic or area-based definition of the natural logarithm ln(a) or ln(b) in the two mutually exclusive cases a > 1 and 0 < b < 1. These definitions imply that ln(x) ® 0 = ln(1) when x ® 1. Reading Guide. The rest of this section states and indicates the proofs of two algebraic properties of the natural logarithm. The first proof is easy. The second proof is cryptic - material for advanced students. The next section briefly indicates the relationship between the inverses of the logarithms and exponential functions - more material for advance students. Consult another calculus or analysis text for the missing details. Proof of Property ln([1/(b)]) = -ln(b) for b > 0.We will show that 0 = ln(b)+ln([1/(b)]) when b > 0. For this, first consider the case a > 1. In the following diagram
By symmetry (or reflection across the line y = s), ln(a) = Area(B)+Area(A). Therefore ln(a) = Area(B)+Area(C)
Now by definition -ln([1/(a)]) = Area(B)+Area(C). Therefore -ln([1/(a)]) = ln(a).This in turn implies ln([1/(a)])+ln(a) = 0.whenever a > 1. Finally, we conclude ln([1/(b)])+ln(b) = 0 whenever b > 0. This follows by putting a = b if b ³ 1 and by putting a = [1/(b)] if 0 < b < 1.) The latter is equivalent to the property ln([1/(b)]) = -ln(b) which we wanted to show. Fundamental Property of LogarithmsNext we may derive the fundamental property of logarithms, that is
Sketch of A First DemonstrationThis indicates a simple demonstration of the fundamental property for the natural logarithm ln(x) for x > 0. The sketch of an alternative proof follows. Sketch of a Second Demonstration. For a > 0, put G(x) = ln(ax). Then value of G(x) is given by the (signed) area from s = 1 to s = ax under the curve y = [1/(s)]. Observe G(1) = ln(a). The area of region D in the following diagram equals G(x+Dx)-G(x). The height of the region D is approximately [1/(ax)] and its length is precisely a(x+Dx) - ax = aDx. Therefore
Logarithms To Base a > 0The logarithm to base a > 0 is given by loga(x) = [(ln(x))/(ln(a))] when a ¹ 1. The property ln(ax) = ln(a)+ln(x) now implies logc(ab) = logc(b) +logc(a) holds when a, b and c are all positive real numbers with c ¹ 1. The proof is a simple algebraic exercise. Further note that ln(e) = 1 implies loge(x) = ln(x).
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Units in Calculations: Enriched material: The Appendices of Volume
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