Semi-Log Graphs

Recently I was introduced to a concept of graphing called semi-log graphing. When most people use graphs, they use linear scales for both the x and y axes, but semi-log graphs use a linear scale for the x axis and a logarithmic scale for the y axis, hence semi-log. How this logarithmic scale works is it goes up from numbers one-ten, then continues as such but with the numbers ten-twenty. Essentially, you ad a zero after the numbers of the scale every ten numbers. This logarithmic scale is very useful for looking at certain functions, like exponential functions. Below are graphs of the functions f(x) = 2^x and g(x) = 3^x on both a semi-log graph and standard graph. You can see the values are much clearer on a semi-log graph, but it can be difficult to recognize the function without understanding the semi-log format.

Standard Graph

Semi-Log Graph

Exponential Functions

Exponential functions are an interesting concept that can be used to describe extreme growth of things and numbers. Whet graphing an exponential function it loos like this: y = n ^ x. This means that every y value is a number and the x value as its exponent. Imagine this; on the internet, somebody posts some information. In the next hour (the internet is very slow today) three people see this information, and in the next hour they each share it with three people, and so on. The numbers of people who see the information greatly increases each hour. By hour fifteen, over ten-million people have seen the information. Now imagine if ten people shared this information every hour (the internet is slightly less broken).  graph of this information on very low levels can be seen below. If everyone could and wanted too, all the people in the world could see the information in less than a day with both these functions. And remember, this is some people sharing every hour, when really information can now be shared every couple seconds. Imagine what would happen if 100 people shared every fifteen seconds? No, the internet needs longer to repair, so thirty seconds. It would take five minutes for more people than those that exist to see the information.

Here you see both these functions, and the points that correspond to the natural numbers that allow visibility. The black line (No, not the y -axis) represents y = 10^ x, and the red line represents y = 3 ^ x. (1, 10) and (2, 100) are the points that represent the black line function. Even at this beginning scale, you can see how quickly the function rises.