Page 160 - Calculus Demystified

CHAPTER 6

Transcendental

Functions

6.0 Introductory Remarks

There are two types of functions: polynomial and transcendental. A polynomial
2
k
of degree k is a function of the form p(x) = a 0 + a 1 x + a 2 x + ··· + a k x .
Such a polynomial has precisely k roots, and there are algorithms that enable us to
solve for those roots. For most purposes, polynomials are the most accessible and
easy-to-understand functions. But there are other functions that are important in
mathematics and physics. These are the transcendental functions.Among this more
sophisticated type of functions are sine, cosine, the other trigonometric functions,
and also the logarithm and the exponential. The present chapter is devoted to the
study of transcendental functions.

6.1 Logarithm Basics

A convenient, intuitive way to think about the logarithm function is as the inverse
to the exponentiation function. Proceeding intuitively, let us consider the function

x
f(x) = 3 .
To operate with this f, we choose an x and take 3 to the power x. For example,

4
f(4) = 3 = 3 · 3 · 3 · 3 = 81
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