# Logarithms/Exponential functions/Introduction/Section

For a positive real number
,
the
*exponential function for the base*
is defined as

Let denote a positive real number. Then the exponential function

- We have for all .
- We have .
- For and , we have .
- For and , we have .
- For , the function is strictly increasing.
- For , the function is strictly decreasing.
- We have for all .
- For , we have .

### Proof

There is another way to introduce the exponential function to base . For a natural number , one takes the th product of with itself as definition for . For a negative integer , one sets . For a positive rational number , one sets

where one has to show that this is independent of the chosen representation as a fraction. For a negative rational number, one takes again the inverse. For an arbitrary real number , one takes a sequence of rational numbers converging to , and defines

For this, one has to show that these limits exist and that they are independent of the chosen rational sequence. For the passage from to , the concept of uniform continuity is crucial.

For a positive real number
, ,
the
*logarithm to base*
of
is defined by

- We have and , this means that the logarithm to Base is the inverse function for the exponential function to base .
- We have .
- We have for .
- We have