In doing the exercises related to this section of the

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In doing the exercises related to this section of the module (and in general), you will need to use the “ y x ” or “ x y ” button on your scientific calculator. Typically, the sequence of commands for, say, “2 3 = ?” is [2 x y 3 =], but if you haven’t used this button before, you should experiment with some simple examples, before doing the exercises. Logarithms and Exponential Functions: If b x = a , then x is the logarithm of a to base b , that is, the power to which b must be raised to give a : log b a = x. The two most common bases for logarithms are 10 and e . Logarithms to these two bases are written “log” (or log 10 ) and ln (for “natural” logarithm), respectively. Recall that e is a transcendental number (with a value of approximately 2.71828) defined as the limit of the expression (1 + 1/ n ) n as n approaches infinity, a formula which represents instantaneous compounding of a growth rate, as n , the number of subperiods within a period, becomes infinite and the growth rate per subperiod becomes infinitesimal. It is possible to convert logarithms from one base to another. If b x = a y , then x log b b = y log b a , and since the logarithm of any number to itself as base (in this case, log b b ) = 1, we have x = y log b a . Your scientific calculator has log 10 , 10 x , ln and e x functions. You may want to confirm by punching 10 into your calculator and hitting the ln button that log e 10 = ln 10 = approximately 2.302585. Hence if log 10 a = x, then log e a = 2.302585 x . Using our knowledge of exponents, we know immediately that log 10 10 = 1, log 10 100 = 2, log 10 1 = 0, and log 10 0.1 = log 10 1/10 = –1. We also know that numbers between 10 and 100 have logarithms between 1 and 2, and we trust our calculators to remember precisely what they are. We shall mention only 2 applications of logarithms and exponents here. First, using a result from Module 10, we have that for the function ln x , the derivative d ( ln x )/ dx = 1/ x, and so d ( ln x ) = dx/x . Since the same relation holds true for a variable
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