Exercises for Chaos Under Control

Chapter 3: The Logarithm Function

Exercises 4-8 should be done with the help of a hand calculator with a "log" button. (Here "log" means log10.)

4.[N] The logarithm is a recipe for converting one number into another. For instance, the number 1 is converted into 0 by the logarithm recipe. Into what numbers are 2, 3, 5, 7, 10, 20, 30, 100, 200, 300, 1000, 2000, 3000 converted? Note that log(2), log(20), log(200), and log(2000) all have the same string of numbers after the decimal point.

5.[N] Note that all numbers can be written as a product of many (actually, infinitely many) factors. For example, 6 = 2*3 = 1.5*4 = 1.2*5, and so on. Verify that log(6) = log(2)+log(3) = log(4)+log(1.5) = log(5)+log(1.2) as required by Equation (1) in the text.

6.[N] The number 9 is the square of 3: 9 = 32. Find log(9) and verify that it is 2*log(3) as required by Equation (2) in the text. Can you write 8 as a number raised to a simple power? Verify that log(8) is that power times the log of the number which is being exponentiated.

7.[N] Now, back to the relation between log(2) and log(2000); 2000 is 2*1000 = 2*103. So log(2000) = log(2*103) = log(2)+3*log(10). But you have seen that log(10) = 1. So log(2000) = 3+log(2). In fact, since 10 = 101 we can say that, in general, the logarithm (to the base 10) of a number is the power to which we would have to raise 10 in order to get the number we started with. If your calculator has a "xy" button do this: verify that log(2) = 0.30102995...; then enter "10"-"xy"-"(3+0.30102995)". The answer should be 2000. Try it for some other numbers.

8.[N] The reciprocal of a number is 1 divided by that number; the reciprocal of 5 is 1/5. Another way of writing reciprocal is to raise the number to the power -1; 1/5=5-1. Find log(5) and log(1/5). (If your calculator calculates log it will almost surely calculate reciprocals; look for the "1/x" button.) Try taking logs of other numbers and comparing them with the logs of their reciprocals. In each case you should observe that log(1/x) = -log(x). This is the special case of Equation (2) above for p = -1. Note that the log of a number less than 1 is always negative. (Try some.)

9. Logarithm mischief. We measured the dimension of the square by noting if we double the side length, we quadruple the mass. The dimension formula, Equation (3), gives

d = log(4)/log(2) = log(22)/log(2) = 2*log(2)/log(2) = 2,

using Equation (2). A "short cut discovered by some beginners" is to "cancel" the logs:

d = log(4)/log(2) = 4/2 = 2.

Of course, this method is incorrect, though it does give the right answer. To convince yourself this method is wrong, use this trick to calculate the dimension of a cube. Can you find another instance in which the method (accidentally) yields a true result?

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