Inverse Functions and ExponentialsThe above geometric definition implies ln(1) = 0. It also implies that ln(2) > 0.5 Note the rectangle of height 0.5 with base segment [1,2] has area 0.5. It also lies strictly beneath the curve u = [1/(v)] where 1 £ v £ 2. Now mathematical induction implies ln(2n) = n ln(2) > n/2 (since 2n+1 = 2n ·2). Now ln(4) = ln(2)+ln(2) > 0.5+0.5 = 1. The continuity
of ln(x) between x = 1 and x = 4 implies by the Intermediate Value Theorem there is at least one number e such that ln(e) = 1. The number value y of the exponential function exp(x) can now be defined as the unique number y satisfying the equation ln(y) = x. This definition of exp(x) leads to the property
Note that the number e is called the natural number. The infinite decimal expansion of e begins with 2.718281828 ¼ Note that the digits 1828 appear twice in this otherwise non-repeating decimal expansion. The number e is irrational. The proof of that e is not rational, is another intellectual mortgage.
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Units in Calculations: Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area. |
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