Logarithms/Exponential functions/Introduction/Section

The exponential functions for various bases


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


Let denote a positive real number. Then the exponential function

fulfills the following properties.
  1. We have for all .
  2. We have .
  3. For and , we have .
  4. For and , we have .
  5. For , the function is strictly increasing.
  6. For , the function is strictly decreasing.
  7. We have for all .
  8. 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

Logarithms for various bases


The logarithms to base

fulfill the following rules.
  1. We have and , this means that the logarithm to Base is the inverse function for the exponential function to base .
  2. We have .
  3. We have for .
  4. We have

Proof