Turning Multiplication into Addition

We are going to use a clever trick to help us study the properties of logarithms. This trick is based on Definition 1 of logarithms. The idea is to use this definition to write the product of x and y in two different ways. Firstly and and so

But we also can treat the product as a combined value to get . Since both of these equal xy we can put them together to get:

To make this useful we must use the first law of indices (when multiplying powers of the same base add the indices). Using this on the RHS we obtain.

From this it seems reasonable to conclude that

A problem with this reasoning

This argument depends on the idea that if then we can depend on the fact that . This is in fact true as you can see from a graph of . This graph is monotonic. This means that it only goes up (monotonic increasing) or down (monotonic decreasing). is monotonic increasing.

Dynamic Graphs Pop up Graph Winplot Sketchpad Help

From this graph we can see that each value of y is connected to only one value of x. All exponential graphs are pretty much the same. If you want to check this out have a look at a dynamic version of the above graph which will allow you to see exponentail graphs with different bases. There is much to investigate with these graphs and we shall return to them. The monotonic property guarantees that if then , and so our above conclusion is justified.

The First Law of Logarithms

The identity

is called the first law of logarithms. It is really just another way of looking at the first law of indices.

A short version of the proof.