483 4.4 Evaluating Logarithms and the Change-of-Base Theorem (b) We use common logarithms for this approximation. log2 0.1 = log 0.1 log 2 ≈ -3.3219 The last two entries in Figure 38(a) show that the results are the same whether natural or common logarithms are used. Some calculators, such as the TI-84 Plus, evaluate these logarithms directly without using the change-of-base theorem. See Figure 38(b). S Now Try Exercises 79 and 81. Check: 2-3.3219 ≈0.1 EXAMPLE 9 Modeling Diversity of Species One measure of the diversity of the species in an ecological community is modeled by the formula H= -3P1 log2 P1 + P2 log2 P2 + g+ Pn log2 Pn4, where P1, P2, c , Pn are the proportions of a sample that belong to each of nspecies found in the sample. (Data from Ludwig, J., and J. Reynolds, Statistical Ecology: A Primer on Methods and Computing, © 1988, John Wiley & Sons, NY.) Find the measure of diversity, to the nearest thousandth, in a community with two species where there are 90 of one species and 10 of the other. SOLUTION There are 100 members in the community, so P1 = 90 100 = 0.9 and P2 = 10 100 = 0.1. H= -30.9 log2 0.9 + 0.1 log2 0.14 Substitute for P1 and P2. In Example 8(b), we found that log2 0.1 ≈ -3.32. Now we find log2 0.9. log2 0.9 = log 0.9 log 2 ≈ -0.152 Change-of-base theorem Now evaluate H. H= -30.9 log2 0.9 + 0.1 log2 0.14 H≈ -30.91-0.1522 + 0.11-3.3224 Substitute approximate values. H≈0.469 Simplify. Verify that H≈0.971 if there are 60 of one species and 40 of the other. As the proportions of n species get closer to 1 n each, the measure of diversity increases to a maximum of log2 n. S Now Try Exercise 73. We saw previously that graphing calculators are capable of fitting exponential curves to data that suggest such behavior. The same is true for logarithmic curves. For example, during the early 2000s on one particular day, interest rates for various U.S. Treasury Securities were as shown in the table. Time 3-mo 6-mo 2-yr 5-yr 10-yr 30-yr Yield 0.83% 0.91% 1.35% 2.46% 3.54% 4.58% Data from U.S. Treasury. Figure 39 shows how a calculator gives the best-fitting natural logarithmic curve for the data, as well as the data points and the graph of this curve. 7 y = 1.479 + 0.809 lnx −2 −5 5 35 Figure 39
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