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Appendix IV: Log-Log Plotting Introduction It is often useful to plot the logarithm of one variable versus the logarithm of another when analyzing data. Such plots allow one to infer whether the relationship between the two variables is of the form y = axn within experimental error. When y = axn, then log y = n log x + log a and a plot of log y versus log x is a straight line with a slope equal to n and an intercept equal to log a. Log-Log graph paper (often called full logarithmic graph paper) has both abscissa and ordinate scales laid out in proportion to the logarithms of the variables. Thus these scales permit plotting log x versus log y directly. There is, of course, no point on the scales representing the zero of either variable. Log - Log paper is available with various numbers of cycles along the axes. If the range of the variables to be plotted spans a factor of 10 or less, i.e., such as 0.1 to 1.0 or 1.0 to 10.0 etc., then 1 × 1 cycle paper is appropriate. If the variables span a range approaching 100 to 1 then 2 × 2 cycle paper should be used, etc. In general, one chooses the number of cycles so that the plots are reasonably spread out over the paper. Log-Log paper is also available with a different number of cycles along the two axes. It must be emphasized that the exponent n in the power law given in the above equation can be obtained directly from measured distances on the log-log graph. Thus the slope can be obtained by choosing the right triangle with most of the fitted straight line as the hypotenuse and using a linear scale, such as ruler to measure the length of the base and the height of the right triangle. The ratio of these measured lengths will then be the slope n. This is true whether or not the log-log paper has the same number of cycles along both axes since in general the paper is made so the individual cycles have the same physical dimension along each axis. One of the disadvantages of the log-log paper is that you may find yourself with paper having the wrong number of cycles for the data in hand. Generally, when the paper has too many cycles, the temptation is to use it anyway. This result is a tiny graph and a larger than necessary error in the experimentally determined slope. This problem can be overcome by using computer programs, which when done properly determine the slope and intercept accurately. Exercise In order to obtain some intuitive feeling for data analysis using log-log plots the following exercise has been prepared to help you understand the mechanics behind log-log plotting. For this exercise you will need 3 × 3 cycle log-log paper. On this paper plot the curve 3 y = 8x 2 for the range of x values from 0.4 ≤ x ≤ 20. Verify by direct measurement that the slope is 3/2 and that the intercept is log 8. One possible solution is shown in the graph on the next page. 71 10 3 y slope = 10 ∆y 180 3 = = ∆x 120 2 2 ∆y = 180 10 1 ∆x = 120 0 10 -1 10 10 0 10 1 x 10 3 Log-Log plot of y = 8x 2 . 72 2