The logarithmic function of base a, where a is positive and not 1, is denoted by (which is read as "y is log base a of x") and is defined by
Domain of logarithmic function = range of exponential function = Range of logarithmic function = domain of exponential function =
In fact the logarithmic function is the inverse of
Properties of the graph
Properties of 1. The domain is and the range is 2. The x-intercept is (1, 0) and there is no y-intercept. 3. The y-axis is a horizontal asymptote 4. is an increasing if and decreasing if 5. one-to-one function 6. The graph contains the three points 7. The graph of f is smooth and continuous. (Here smooth means you can take as many derivatives as you want)
Sometimes a logarithm function is written without making reference to a base, for example
When this happens the base is assumed to be 10. This means
There is a special base, e, to which we associate a special logarithm , which is called the natural logarithm.
Notice that we do not write the base. That is whenever we use the natural logarithm, we are using base e.
Note: e is about 2.71828...