Power, Exponential, and Logarithmic Functions



The basic power function is

y = xn
where n is a positive integer. You know how this can be extended by algebra to define
y = xb
when b is a fraction or a negative integer.

Bring up a Maple window and plot the functions

y = x4,        y = x1/4 = 4th root of x,        y = x-4.
Keep Maple open and use it while you work through the rest of this document. (But remember to kill your old plot windows frequently if you are using Release 3.)

It is possible (we skip the logical details for now) to ``fill in'' the definition to make sense of xb for any real number b (at least if x is positive).

Plot

y = x3.9,        y = x.24,        y = x-4.1,
each on the same axes as the corresponding function above. Experiment with other values of the exponent.

The equation

ab = c
defines several functions, depending on which of the 3 quantities is the dependent variable, which the independent variable, and which just a constant. (Note: To avoid complications, we assume for the rest of this discussion that a>0 and a ¹ 1.)

We already talked about the cases y = xb (power functions) and

yb = x   =>    y = x1/b = bth root of x
(root functions, which are just additional power functions). Our primary interest today is the more exotic cases:

Exponential functions:     y = ax

Plot

y = 3x,        y = (0.5)x,        y = 1x.
Experiment with other values of the base (a).

Logarithmic functions:

ay = x   =>    y = loga (x)

Plot

y = log3 (x),        y = log(0.5) (x).
(Don't confuse log3(x) with log(3x). The Maple syntax is log[3](x).) Experiment with other values of the base. (Why is the case a = 1 pathological?)

The most important fact to memorize about exponential and logarithmic functions is that most of those functions are unnecessary to memorize about!

Theorem: There is a number e » 2.718 ¼ such that

ax = ex ln(a)        and        loga (x) = ln(x)
ln(a)
  ,
where  ln  is defined by
ln(x) = loge (x).

Therefore, exponential and logarithmic functions with respect to an arbitrary base a can be eliminated in favor of those with respect to the special base, e.

Plot

y = ex,        y = ln x,        y = 2x,        y = log10 x.
(Note that Maple likes you to write exp(x), not e^x.)

Notational remarks: ex is also written as  exp x. Then  exp  and  ln  are thought of as new transcendental functions like  sin  and  cos . Just as for the trig functions, the parentheses are often left off the arguments of these functions when there is no ambiguity:

exp x ,        log10 3,        ln x2.
(Finally, pure mathematicians write  ln x as  log x, but engineers and scientists don't like that.)

Properties of the graphs


The function ex increases faster at infinity than any power function.

Plot y = ex and y = x3 on the same axes. Then plot y = ex and y = x8 on the same axes. Do you still believe the statement? Make the x scale big and the y scale huge!

The function e-x decreases faster at infinity than any negative power.

Plot y = e-x and y = x-2 on the same axes. Then plot y = e-x and y = x-20 on the same axes; experiment with the scales to find the crossover point.

The function  ln x increases more slowly at infinity than any positive (fractional) power.

Plot y = ln x and y = x1/5 on the same axes. Make the x scale bigger until you find the crossover point.

As x approaches 0, the function - ln x increases more slowly than any negative power.

Plot y = - ln x and y = x-1/5 on the same axes. Do you believe the statement?

Algebraic properties of exponentials (``the laws of exponents'')


ex+y = ex ey
(ex)y = exy
e-x = 1
ex
(ab)x = ax bx
ex >= 0
e0 = 1
e1 = e ;        a1 = a
Except as noted in the last line, these laws also hold for ax.

Algebraic properties of logarithms


ln(xy) = ln x + ln y
ln(xy) = y ln x
ln(1/x) = - ln x
ln x is defined only for x>0.
ln 1 = 0 ;        ln x < 0 if and only if 0<x<1.
ln e = 1 ;        loga (a) = 1
Except as noted in the last line, these laws also hold for  loga .

File translated from TEX by TTH, version 0.9 (then tweaked by hand by S.A.F.).