Numbers, bases and logarithms Take a number, say 565937. This number has an existence of its own independent of the numbering system used to represent it. The mathematics of communication addresses the problem of determining the cost of recording or transmitting this number and, beyond numbers, any conceivable message. Staying with 565937, suppose we used for our numbering system a base different from 10. The trade-off between size of base and length of representation manifests what is called the amount of information carried by the number.
Information always depends on the context in which a signal is to be understood.
In many cases, however, it is not practical to make up a different encoding system for each context. For example, in encoding numbers, one usually makes the hypothesis that in each place all digits are equally likely. So in writing a six-digit decimal number we are choosing one of the one million possibilities. To get a standard measure of this information, and for practical reasons as well, we use base 2. Here is where logarithms come into the picture.