Power, Exponential, and Logarithmic Functions

The basic power function is

y = x^{n}

where n is a positive integer. You know how this can be extended by algebra to define

y = x^{b}

when b is a fraction or a negative integer.

Bring up a Maple window and plot the functions

y = x^{4},  y = x^{1/4} = ^{4} \sqrt{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 x^{b} for any real number b (at least if x is positive).

Plot

y = x^{3.9},  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

a^{b} = 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.)

We already talked about the cases y = x^{b} (power functions) and

y^{b} = x ⇒ y = x^{1/b} = ^{b} \sqrt{x}

(root functions, which are just additional power functions). Our primary interest today is the more exotic cases:

Exponential functions: y = a^{x} Plot

y = 3^{x},  y = (0.5)^{x},  y = 1^{x}.

Experiment with other values of the base (a).

Logarithmic functions:

a^{y} = x ⇒ y = log_{a}(x)

Plot

y = log_{3}(x), y = log_{(0.5)}(x).

(Don’t confuse log_{3}(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 … such that

a^{x} = e^{x ln(a)}  and  log_{a}(x) = \frac{ln(x)}{ln(a)} ,

where ln is defined by

ln(x) = log_{e}(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 = e^{x},  y = ln x,  y = 2^{x},  y = log_{10} x.

(Note that Maple likes you to write exp(x), not e^x.)

Notational remarks: e^{x} 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,  log_{10} 3,  ln x^{2}.

(Finally, pure mathematicians write ln x as log x, but engineers and scientists don’t like that.)