General Logarithmic and Exponential Functions
We close this section by looking at exponential functions and logarithms with bases other than
Exponential functions are functions of the form Note that unless we still do not have a mathematically rigorous definition of these functions for irrational exponents.
Let’s rectify that here by defining the function in terms of the exponential function
definition of general exponential functions
For any and for any real number define as follows:
Now is defined rigorously for all values of .
This definition also allows us to generalize property iv. of logarithms and property iii. of exponential functions to apply to both rational and irrational values of It is straightforward to show that properties of exponents hold for general exponential functions defined in this way.
Let’s now apply this definition to calculate a differentiation formula for We have
The corresponding integration formula follows immediately.
derivatives and integrals involving general exponential functions
Let Then,
and
If then the function is one-to-one and has a well-defined inverse. Its inverse is denoted by Then,
Note that general logarithm functions can be written in terms of the natural logarithm. Let Then, Taking the natural logarithm of both sides of this second equation, we get
Thus, we see that all logarithmic functions are constant multiples of one another. Next, we use this formula to find a differentiation formula for a logarithm with base Again, let Then,
derivatives of general logarithm functions
Let Then,
Evaluate the following derivatives:
Evaluate the following integral: