Powers and Logarithms

What they are and how to work with them

Steve Sugden

Bond University

31 August 2011

Index laws and powers

Large numbers

Suppose we want to multiply many copies of the same number together.

Examples

1, 000, 000 = 10 10 10 10 10 10 1, 000, 000, 000 = 10 10 10 10 10 10 10 10 10 1, 000, 000, 000, 000 =

10 10 10 10 10 10 10 10 10 10 10 10

It gets very tedious (not to mention error-prone) to write such numbers.

We ran out of space on the last line!

We need a more compact notation, called index notation. These are used to represent iterated multiplication of the same number by itself many times.

Index laws and powers

We use an index (also called power or exponent or logarithm) to indicate many copies of the same number multiplied together.

Examples

1, 000, 000 = 106 1, 000, 000, 000 = 109

1, 000, 000, 000, 000 = 1012

In general, for any number x, and an integer n 2, we write xn to mean n copies of x multiplied together.

First index law

Now that we have a new notation, we need to see how it behaves, i.e., what the rules are for this new notation. To get a feeling for what is going on, we …rst consider some examples.

Example

Suppose we have three copies of 2 multiplied together. Then multiply that by another four copies of 2. Surely this gives us seven copies altogether.

23 24 = (2 2 2) (2 2 2 2)

=2 2 2 2 2 2 2

=27

First index law

Example

More generally, take m copies of 2 multiplied together then multiply that by another n copies of 2 to give us m + n copies altogether.

2m 2n = 2m+n

Fact

Even more generally, take m copies of x multiplied together then multiply that by another n copies of x to give us m + n copies altogether.

xm xn = xm+n

In these examples, m 2 and n 2 and both are integers. We can extend to m 1 and n 1 by de…ning x1 = x.