Graphing Logarithmic functions

Introduction to Logarithm functions : If a > 0 and and a is not equal to 1, then the function is defined by f(x) = log base a x, x > 0 is called the logarithmic function. We have learnt that the logarithmic function and the exponential function are inverse functions i.e., log base a x = y => x = a^y. We observe that the domain of the logarithmic function is the set of all non-negative real numbers i.e. (0, infinity) and the range is the set R of all real numbers. As a > 0 and a is not equal to 1. So, we have the following cases.let us understand Graphing Logarithmic Functions.

To Graph Logarithmic Functions we will consider two cases:

Case I: When a > 1, In this case, we have y = log base a x = < 0 for 0 < x < 1=> = 0 for x = 1= > 0 for x > 1 , Also the values of y increase with the increase in x.

Case II: When 0 < a < 1, In this case, we have y = log base a x = > 0 for 0 < x < 1=> 0 for x = 1=> 0 for x > 1. Also the values of y decrease with the increase in x. Let us take some example  to Graph Logarithmic Functions

Example 1: Graph the function y = |log base e |x||.
Solution: In order to draw Logarithmic Function Graph  of y = |log base e |x||, we follow the following steps:
Step 1: Draw the graph of y = log base e x for x > 0 to obtain the graph of f(x) = log base e |x| for x > 0.
Step 2: Reflect the graph obtained in step 1 about y-axis to obtain the graph of y = log base e |x| for x < 0.
Step 3: Keeping the portion lying above x-axis of the graph unchanged reflect the portion lying below x-axis about x-axis. This will give the graph of y = |log base e |x||.

Example 2: Graph the function y = x^2 – 2 |x|.
Solution: We have, y = x^2 – 2 |x|
y = |x|^2 – 2 |x|
y = f(|x|), where f(x) = x^2 – 2x.
Thus, to draw Logarithmic Function Graph  of y = x^2 – 2 |x|, we follow the following steps:
Step 1: Draw the graph of y = x^2 – 2x for x greater than equal to 0.
Step 2: Reflect the graph obtained in step 1 about y-axis to obtain the graph of y = x^2 – 2x for x < 0.