A logarithmic (or just “log”) scale has unevenly spaced grid lines. A standard scale has evenly spaced grid lines. Some data needs to be graphed on standard paper only, some on semi-log graphs, and some on log-log graphs. For example, the graph of y=x{\displaystyle y={\sqrt {x}}} (or any similar function with a radical term) can be graphed on a purely standard graph, a semi-log graph, or log-log graph. On a standard graph, the function appears as a sideways parabola, but the detail for very small numbers is difficult to see. On the log-log graph, the same function appears as a straight line, and the values are more spread out for better detail. [3] X Research source If both variables in a study include great ranges of data, you would probably use a log-log graph. Studies of evolutionary effects, for example, may be measured in thousands or millions of years and might choose a logarithmic scale for the x-axis. Depending on the item being measured, a log-log scale may be necessary.
e{\displaystyle e} is a mathematical constant that is useful in working with compound interest and other advanced calculations. It is approximately equal to 2. 718. [4] X Research source This article will focus on the base-10 logarithms, but the reading the natural logarithm scale operates in the same way. Standard logarithms use base 10. Instead of counting 1, 2, 3, 4… or 10, 20, 30, 40… or some other evenly spaced scale, a logarithm scale counts by powers of 10. The main axis points are, therefore, 101,102,103,104{\displaystyle 10^{1},10^{2},10^{3},10^{4}} and so on. [5] X Research source Each of the main divisions, usually noted on log paper with a darker line, is called a “cycle. ” When specifically using based 10, you can use the term “decade” because it refers to a new power of 10.
The minor interval marks are based on the logarithm of each number. Therefore, if 10 is represented as the first major mark on the scale, and 100 is the second, the other numbers fall in between as follows: log(10)=1{\displaystyle log(10)=1} log(20)=1. 3{\displaystyle log(20)=1. 3} log(30)=1. 48{\displaystyle log(30)=1. 48} log(40)=1. 60{\displaystyle log(40)=1. 60} log(50)=1. 70{\displaystyle log(50)=1. 70} log(60)=1. 78{\displaystyle log(60)=1. 78} log(70)=1. 85{\displaystyle log(70)=1. 85} log(80)=1. 90{\displaystyle log(80)=1. 90} log(90)=1. 95{\displaystyle log(90)=1. 95} log(100)=2. 00{\displaystyle log(100)=2. 00} At higher powers of 10, the minor intervals are spaced in the same ratios. Thus, the spacing between 10, 20, 30… looks like the spacing between 100, 200, 300… or 1000, 2000, 3000….
Date Time Age Medication given
Population growth rates Product consumption rates Compounding interest
For example, the number 4,000,000 would be graphed at the fourth minor scale mark above 106{\displaystyle 10^{6}}. Even though, on a standard linear scale, 4,000,000 is less than halfway between 1,000,000 and 10,000,000, because of the logarithmic scale, it actually appears slightly more than halfway. You should note that the higher intervals, closer to the upper limit, become squeezed together. This is due to the mathematical nature of the logarithmic scale.
