Sixth index law

We may extend these ideas still further to power 1/2, such as 21/2.

Again, we require the laws already established to be respected.

We may then attach a meaning to power 1/2 as follows.

Example

If expressions involving the power 1/2 are to obey the …rst law, then, for example, 21/2 21/2 = 21/2+1/2 = 21 = 2. So we have 21/2 2 = 2.

Example

More generally, x1/2 x1/2 = x1/2+1/2 = x1 = x

Fact

Since x1p/2 x1/2 = x, the only reasonable de…nition for x1/2 is that x1/2 = x.

Logarithms

A logarithm is nothing more than a power, just like we have been working with already.

Why then do we need this new word?

Logarithms are used when the focus of a relationship is on the power.

For example, we know that 23 = 8. So if we want to highlight the fact that the power we use to raise 2 to get 8 is 3, we say that the logarithm of 8 is 3 (to base 2). We write:

23 = 8

log2 8 = 3

Fact

If bL = x then L = logb x

Logarithm properties

Corresponding to every index law, there is a logarithm law.

This should not be surprising, because an index is a logarithm.

In each of the following, we have b > 0.