Logarithmic Functions

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Logarithmic Function

 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)

Common Logarithm

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

Natural Logarithm

 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...

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