How to Sketch the Graphs of Logarithm functions, an Easy Way

The Graphs of Logarithm functions can be easily sketched by using three points on the Y-Axis and three points on the X-Axis. The points on the Y-Axis are, Y=1, Y=0, and Y=-1. To determine the points on the X-Axis, we use the base of the logarithm. If the Base of the logarithm is the number 'b', where b > 0 and b ≠ 1, then the points on the X-Axis, that corresponds respectively with the points on the Y-Axis,
are; x=b, x=1, and x=(1/b). The coordinates of the points that the graph would pass through are (1/b,-1),(1,0) and (b,1). When working with these points, the graphs can be easily sketched.

Things You'll Need

• Paper and
• Pencil
• Graph Paper

Instructions

1. 

   The Graphs of Logarithm functions can be easily sketched by using three points on the Y-Axis and three points on the X-Axis. These three points will be Y=1, Y=0, and Y=-1. To determine the points on the X-Axis, we use the base of the logarithm. For example, let us graph the function f(x)= logx, where the base of this log is 10 and the Argument is 'x'. When the Argument of the log is equal to the Base of the log, then Y=1. When the argument of the log is 1 to any base, then Y=0. If the argument of the log is the reciprocal of the base, then Y=-1. The coordinates of the points that this graph would pass through are (1/10,-1),(1,0) and (10,1).Please click on the image to get a better understanding.

2. 

   To Sketch the graph of the log of X to base 7, we choose the three points on the X-axis, X=1/7, X=1, X=7. On the Y-axis, Y=-1, Y=0, Y=1. So the three points that the graph will pass through is (1/7, -1), (1, 0), and (7, 1). Please click on the image to see the graph.

3. 

   To graph the log of X to base (1/5), we mark of on the X-axis, X=1/5, X=1, and X=5. On the Y-axis, Y=1, Y=0, and Y=-1. So the three points that the graph will pass through is (1/5, 1), (1,0), and (5, -1). Please click on the image to see the graph.

4. 

   To graph the log of X to base e, that is Y=Ln x, we mark of on the X-axis, X=1/e, X=1, and X=e where e is approximately 2.781828182846... On the Y-axis, the corresponding points will be Y=-1, Y=0, and Y=1. The points are (1/e, -1), (1, 0), and (e, 1). Please click on the image to see the graph.

5. In general, to graph the log of X to base b where b>0, and b≠1, the X-axis will have points 1/b, 1, and b. On the Y-axis, the corresponding points will be -1, 0, and 1. So the points the graph will pass through will have the coordinates
   (1/b,-1), ( 1, 0) and (b, 1).