which in fact converges .
We have assumed that we can multiply infinite series as if they were finite algebraic expressions. This is not always true, but it is for convergent power series.
The relationship leads us to write so that .
On the assumption that a power series can be differentiated term by term .
Properties of exp
The graph of is as shown, (with series definition this is not easy to see), but there is some evidence from the property
- is continuous everywhere.
- is an injection (or one to one).
This means there will be an inverse function. We call it the log function. We have equal to the unique x such that .
Properties of ln
- If then , so .
- is an injection.
Put and .
So, and .
Whence by 2.
(i) if m in an integer then .
(ii) if is in , then equally we have . (.
For and we define .
For example .
Note that this definition fits in with the usual one when x is rational.
For fixed is a function.