Formal Definitions

We developed a meaning for logarithms on the previous page Now we will summarize this in the following three different ways. Each definition will give us a different viewpoint and together they will give us some powerful tools.

Definition 1: Logarithms are indices.

If we try to write a number x as a power of a then the required index would be called the base a logarithm of x. So

This definition gives us a useful simplification formula, as well as a neat way of rewriting x to study the properties of logarithms.

Definition 2: Logarithms solve exponential equations.

The solution of is .

This process works in reverse:

The solution of is .

The two equations and are completely equivalent. They are different ways of saying exactly the same thing. They show that logarithms can be used to solve exponential equations. Also when the need arises later we can use exponentials to solve logarithm equations.

Definition 3: Logarithms undo exponentials.

If you take a logarithm of an exponential you get back what you started with. So

This is also a useful simplification formula.

Logarithms and Exponentials undo each other.

Let's look at some order of operations in these definitions. On the RHS of definition 1, starting with x, we do a base a logarithm followed by a base a exponential. The result is x since these undo each other.

In definition 3 it's the other way around. Starting with y we firstly do a base a exponential followed by a base a logarithm. The result is y since these again undo each other.

This is what it looks like in a diagram. Firstly with numbers ...

and also with algebra.

We will use these important ideas to investigate the properties of logarithms.