We can't have logarithms without bases.
If
then
and we say we are 'finding log base
of
'.
For example if
then
to 3 decimal places.
Some bases are more commonly used than others. Base 10 is the most commonly used. This is because we have 10 fingers, so the most commonly used number system is in base 10. Some cultures use other bases. There is a tribe in West Africa that uses base 20 – 20 fingers and toes – and our time system uses base 60: 60 minutes in a hour, 60 seconds in a minute.
In maths though, the most natural base is
This number arises naturally because the solution to the differential equation
is
In some senses the base is not really relevant since an exponential can be expressed in any base. If
then we can write
and the base is then 2.
Bases less than one are also possible. If
then the base is
We can write
and
so
All bases have in common though that they must be positive, reflecting that the equationhas no solution in general if
Exponential functions with different bases are shown in the graph below.
