We define

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 .

Note,

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

Observe from the graph the properties:

- .
- .
- .
- is continuous everywhere.
- .
- is an injection (or one to one).

ie. .

This means there will be an inverse function. We call it the log function. We have equal to the unique x such that .

and mean of course the same thing. The graph of will be a reflection of in the line .

Note that the domain of ln = range of exp = .

##### Properties of ln

- If then , so .

Therefore, . - is an injection.

**Proof**:

Let .

Put and .

So, and .

Whence . - .

**Proof**:

Let .

Then .

Whence by 2.

So, . - .

**Proof**:

.

(i) if m in an integer then .

(ii) if is in , then equally we have . (.

##### Definition

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.

Note: .