An Unusual Exponential Graph

Abstract

T his paper is an addition to the series of papers on the exponential function begun by Albert Bartlett. 1 In particular, we ask how the graph of the exponential function y = e -t/t would appear if y were plotted versus ln t rather than the normal practice of plotting ln y versus t. In answering this question, we find a new way to interpret the mean life (or time constant) t using such a linear-log graph.Physics teachers are familiar with the graph of the exponential function. For example, the number of radioactive nuclei as a function of time N(t) is given by the following equation, where N 0 is the original number of nuclei.(1) Figure 1 shows the graph of versus t. We have used a mean life t of 4.0 s. We express this time as 4000 ms because it makes comparison of all the graphs presented in this paper easier.A physicist is often interested in the mean life τ (= ln 2 3 the half-life). To give τ a geometrical meaning, plot the natural logarithm of versus t as shown in Fig. 2.This graph, described by Eq. (2), is a straight line whose slope is .(2)In this case, the mean life τ is 4000 ms (or 4.0 s), so the slope of the graph in Fig. 2 is -0.25 s -1 . We want our students to develop the habit of asking "what if…?" questions (for example, see Ref.2). In this spirit, ask the following: "What shape would we find if we plotted versus ln t rather than ln versus t?" Figure 3 is such a graph.We can estimate the mean life from the preceding three graphs with varying degrees of difficulty. Using the linear plot in Fig. 1, the mean life is the elapsed time required for the ordinate to decrease to approximately 1/3 of its original value. (One can also see this on the log-linear graph in Fig. 2, but care must be exercised when interpolating on a logarithmic scale.) In Fig. 2, the mean life is the reciprocal of the magnitude of the log-linear graph's slope; one cannot tell at a glance what the slope is, let alone its reciprocal. In contrast, the unique and most interesting feature of Fig. 3 is that one can see the mean life at a glance: it is the time at which the graph's concavity changes. For the reader's convenience, the mathematics behind this change of concavity at the mean life is presented below using From elementary calculus we know that the concavity of the graph of y versus ln t will change when the second derivaExponential Decay

Why Was Kelvin's Estimate of the Earth's Age Wrong?

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