A logarithmic function relates the output corresponding to the logarithms of the variables. In many applications in real life, it is easier to study the output to the logarithmic values of the variable rather than the values of the variables. This is because; in certain cases the output increases very rapidly with even nominal increases of values of the variables.
Since the logarithm picks up only the increase in the exponent size, the increase rate is controlled to smaller figures. For example if a function is described as f(x) = x, for every1 increase of the variable, the function rises by 1. On the other hand if we define a function g(x) = log [(f(x)] = log (x), then the function rises by 1 only for every 1 increase of exponent of 10.(Here as per normal conventions we referred ‘log’ to mean logarithm to base 10).
In other words, a logarithmic-function has a low increase for large inputs. But however much we explain, the concept can be more clearly understood with a logarithmic function graph. Always a visual presentation is much more effective than verbal algebraic explanation. Thus graphing logarithmic functions have taken a very important place in algebra especially in the study of functions.
Because of the large variations of scales of the input and output in such functions and also because of the domain restrictions one must adopt the correct technique to graph logarithmic functions.
The following may be used as some tips to do the graphing for functions of logarithm.
1) First find the domain of the given functions. Remember that logarithms are defined only for positive expressions. So accordingly determine the domain and start graphing only from the minimum domain value. This will give an idea about the scaling the variable axis.
2) According to the domain of the function, determine the range. You can determine a practical range so that graph is properly sized. This will help in selecting proper scale for the output or function axis.
3) Next find the x-intercept and y-intercept. In some cases one or even both may not exist at least in the selected domain and range of the graph. Plot these points on the set of coordinate axes.
4) Make a table of values for a few points (practically it will be only 2 or 3) selecting compatible values of the variables. For example, if f(x) = log (x), make the table for values of powers of 10. Plot the table values on the same set of axes.
5) Draw a smooth curve connecting all points that are plotted. The ends of the curve should have arrow heads in the appropriate directions to indicate that the function is continuous.