M1.06: Logarithmic Graphs Part I

Section 3: Logarithmic graphs

Data that follows an exponential model is particularly well suited to logarithmic compression, because in that case the graph of the logarithms of the output values will form a straight line. Also, exponential data that covers several half-lives or doubling times will be more easily examined in the logarithmic form.

The display advantages of the logarithmic graph can be combined with the convenience in the original graph of having the scale on the left in the same form as the data. This is done with semi-log graphs, in which the logarithmic graph is displayed but the numbers shown in the y scale on the left are original, pre-logarithm values. The horizontal lines and numbers labeling them are placed at the vertical position that matches the corresponding logarithm.

 graph1 graph2 graph3

To make a semi-log graph, first make a regular graph, then format the y axis:

  1. click on the vertical scale to select it,
  2. use Format > Selected Axis to display the Format Axis dialog box,
  3. click on the Scale tab to show the axis settings,
  4. put a check in the Logarithmic Scale box near the bottom.

Example 5: Make semi-log graphs of these three datasets.

graph4graph5

graph6

x y
1 4.1
2 5.5
3 7.4
4 10.0
5 13.5
6 18.2
7 24.5
8 33.1
x y
10 2
20 5
30 15
40 47
50 145
60 449
70 1,394
80 4,329
x y
-20 67.9
-10 45.1
0 30.0
10 19.9
20 13.3
30 8.8
40 5.9
50 3.9

 

Example 6: For this exponential dataset, make a graph of the logarithms of the y data values and use that graph to estimate what y value is to be expected for an x value of 65.

x y
0 26
10 69
20 181
30 482
40 1,279
50 3,393
60 9,002
70 23,885
Example 7: One of these datasets has an exponential-growth pattern, while the other has a quadratic pattern (i.e., part of a parabola) whose shape is different but close enough that it is difficult to tell just by looking at the graphs. Use graphs of the logarithms of the y values to identify which dataset is exponential.

Dataset A
x y
0 3.5
2 4.1
4 6.1
6 9.3
8 13.7
10 19.5
12 26.5
14 34.9
16 44.5
18 55.3
20 67.5
22 80.9
Dataset B
x y
0 3.6
2 4.8
4 6.4
6 8.4
8 11.1
10 14.7
12 19.5
14 25.8
16 34.1
18 45.0
20 59.6
22 78.8