The Graph of a Logarithmic Function
The graph of 
Domain and Range
For b>1, we may use any positive real number as the argument of 
. Therefore, the domain of 
is 
. As we’ve observed, any real number can be the output of a logarithmic function. Thus the range of f(x) is all real numbers.
Intercepts
For any b>1, 
. The graph has an x-intercept at (1, 0). Since the function is not defined at zero, there is no y-intercept.
Asymptotes
The range of this continuous function is all real numbers; there are no horizontal asymptotes. As x approaches 0, the value of f(x) approaches negative infinity. The line x = 0 is a vertical asymptote of the function. (Note that the function approaches positive infinity as x approaches positive infinity.)
Shape of the graph
The function 
is continuous on its domain. Further, the function is strictly increasing; thus it contains no maximum or minimum points. The graph will always contain the points (b, 1) and 
. With these, you should be able to create a reasonable sketch of the graph.
When 0<b<1
The domain and range of the base-b logarithm do not change, but the function is strictly decreasing in this case. As x approaches 0, f(x) approaches positive infinity. As x approaches positive infinity, f(x) approaches negative infinity. The line x=0 is still a vertical asymptote of the function; all other properties are also unchanged. Note that the graph of y=logax is an “x-axis reflection” of the graph of y=logbx if a and b are reciprocals.
