Algebra/Chapter 12

1.25rem Chapter 12: Exponents and Logarithms Chapter 11 Chapter 13 Algebra Functions using exponents: For positive a and variable x, two functions arise: xa and ax. The first generalizes , the monomials. It also generalizes for natural numbers p and q, so that a is a rational number. The meaning of xa for irrational numbers can be given by the limit of a sequence of rational numbers that converges to a. Now consider the alternative ax where a is constant and x is variable. These functions were presented by Leonard Euler in 1748 to clarify the logarithm concept. This teaching aid has been widely adopted as logarithms are in the realm of transcendental functions, not given by an algebraic formula. The sections below contain both the Eulerian development via the inverse exponential functions ax, and a generation of the natural logarithm as areas bounded by y = 1/x. The shape of such an area may be a hyperbolic sector or a four-sided figure against an asymptote. Sections Exponential Growth and Decay Introduction to Logarithms Natural Logarithm via Hyperbolic Angle