How to Convert to a Log Scale

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A logarithmic scale is used to graph exponential curves so that detail of small units can be shown in the same graph as the general curve shape at large units. The decibel scale, Richter scale, pH scale, and stellar magnitude of brightness are examples of uses of logs to fit data points of an exponential function into a single graphical representation. The general strategy is simply to take the logarithm of the data point values. Note that the data points have to be positive in order to take the log. The most common base for such logarithms is base-10.

Things You'll Need

  • Calculator that computes logarithms
  • Graph paper
  • Logarithmic graph paper (optional)
  • Take the base-10 logarithm of the exponentially behaving coordinates of your data points.

    For instance, if your data points (x,y) behave logarithmically for x, then solve for (log x,y). For example, if your first two data points are (20,3) and (180,5), then convert them to (1.301,3) and (2.255,5). Note that a calculator will tend to have two logarithm buttons: the natural log, and the base-10 log. The former button is usually designated "ln" and the latter reads "log." Usually, log scales are written in base-10, as a convention. The ln button is usually reserved for those natural processes which are characterized by continuous rates of growth.

  • Graph your new data points on regular (not logarithmic) graph paper.

    Since you have already converted to a log scale, you don't need logarithmic graph paper to do it for you.

  • Alternatively, graph the original data points (before calculating their logs) to logarithmic graph paper.

    Note that for units in the tens, increments on the graph paper are in tens; for units in the hundreds, increments on the graph are in hundreds; and so on. Log paper makes a smooth transition between scales, i.e., the increments get narrower as one approaches 10, then narrower again as one approaches 100, and so on.

  • Note that if both terms of the data points are exponential, they will both need logarithmic conversion so that detail at small units can be seen. Therefore, either you can take the log of both terms or you can use what is called log-log graph paper, in which both axes increment logarithmically.

References

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