THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS

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Exponents. The process of defining the meaning of aⁿ is a multi-step process.

Step 1. Define the meaning of the symbol aⁿ where a is any real number and n is a positive integer as follows:

a¹ = a

a² = a • a

a³ = a • a • a

.....

aⁿ = a • a • a .... to n factors

This gives us a meaning for aⁿ when the exponent n is a positive integer. The following laws are valid for aⁿ defined as above (m and n are positive integers):

Next we want to give a meaning to such symbols as

Our object will be to define these symbols in such a way that laws 1 through 5 will apply in all cases.

Step 2. Assign a meaning to a⁰. If we want Law 1 to hold so that

we must assign the value one to the symbol a⁰.

Def. a⁰ = 1.

Step 3. Assign a meaning to negative integral exponents. If we wish Law 1 to hold so that

a^m • a ^{- m} = a^{m - m} = a⁰ = 1 (m is an integer, a≠0)

we must agree that a ^{- m} = 1/a^m.

Def. a^-m = 1/a^m where m is an integer.

Def. n-th root. If aⁿ = p, where n is a positive integer, we call a an n-th root of p, written There may be more than one real n-th root of p. For example since 2² = 4 and (-2)² = 4, there are two real square roots of 4, namely 2 and -2.

Step 4. Assign a meaning to fractional exponents. If we wish Law 1 to hold so that

we must define the symbol a^½ to stand for a square root of a. To avoid ambiguity we can define it to stand for the positive square root. In general, we define the symbol

to stand for the positive q-th root of a.

It follows from these definitions that

and that the above five laws are valid if m and n are either fractions or integers (i.e. if they are rational numbers), provided a and b are positive numbers. If the exponent is irrational, the power is defined to be the quantity approximated by using rational exponents which approximate the irrational exponent; e.g. 3 with exponent denotes the limit of the sequence