1: Logs of graphs and Graphs of logs

Learning Outcomes

After completing this module you should be able to:

  • Explain how log-transformation turns graphs of exponential growth into straight lines.
  • Explain log-transformation of data into graphs by two methods:
    • Log then graph: take the log of the data and then graph the transformed data on normal graph paper or,
    • Graph then log: graph the untransformed data on semi-log paper and let the paper “take the log” of the data.

Logs of graphs and Graphs of logs

In this module we’re going to discuss ways of combining logs and graphics. You remember logs,
right? Just in case you don’t, here are two pictures of logs again:

A size 5 log

log(100,000) = 5

A size 9 log

log(1,000,000, 000) = 9

log(100,000) = 5 log(1,000,000,000) = 9


Remember that the log of the measurement basically tells you how much space that number takes up.

If the measurement is bigger than one, it “takes up space” BEFORE the decimal point and the log is positive. A 6-digit number has a log of 5-point-something. A 10-digit number has a log of 9-point-something.

If the measurement is smaller than one, it only “takes up space”
AFTER the decimal point, and the log is negative.

A measurement of zero doesn’t take up any space before OR after the decimal
point, and it has no log. Neither does a negative measurement. That’s okay,
because usually we’re using logs to measure things like size, weight, height, or the number of organisms in
a population. These measurements can’t be negative in any case. (For more details about logs, click here.)

It’s all very well to take logs of individual numbers, but what we really want is to be able to visualise
these numbers in relation to other numbers. Therefore we need to make graphs. There are two ways you can do this.

  1. Take the log of each number, and then make a normal graph.
  2. Make a graph, and let the paper “take the log” of the number.

You guessed it, we’re going to look at each of these options below.