Logarithmic Function

A basic logarithmic function has a form of f (x) = log_base (x), where x > 0 and the base is positive. Usually, two types of logarithmic functions are used:

a) logarithm of base 10 : log (x) or lg (x)
b) logarithm of base e (natural logarithm): ln (x)

The red curve given below is of base 10 by default and it is extended to the form of f(x) = a log_b (cx + d) + e in order for you to shift, scale or reflect the graph; while the blue curve is of base e (the natural logarithm), which the function is g(x) = a ln (cx + d) + e. You may try to change the base (the value of b) of the red curve to e, which is approximately 2.7 and compare with the blue one.



Slide the values of a, b, c, d and e to see the changes of the graph. You may find out that as c > 0 AND

1. a > 0, the graph is increasing;
2. a = 0, the graph is constant (which is not considered as an exponential function)
3. a < 0, the graph is decreasing.

What other features that you can find for logarithmic function?

Logarithmic function is actually the inverse of exponential function. Graphically, inverse reflects at each other through the line y = x. The graph given below is the form of f(x) = a log (bx+c) + d (base 10). Can you find the general form of exponential function given in blue dotted curve? You may check your answer by clicking the checkbox at the bottom right corner of the graph below.


Betty, Created with GeoGebra