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11Asn3 - ucsc supplementary notes ams/econ 11a Exponential...

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ucsc supplementary notes ams/econ 11a Exponential and logarithm functions c 2010 Yonatan Katznelson The material in this supplement is assumed to be mostly review material. If you have never studied exponential and/or logarithm functions before then you should find a text that covers this material in greater detail (e.g., the full version of our textbook). 1. Exponential functions Definition 1. A function of the form y = b x , where x is the independent variable and b is a constant, is called an exponential function. Comments: The constant b that appears in the definition above is called the base of the expo- nential function. In order that the exponential function be defined for all (real) x the base is assumed to satisfy b > 0. We also assume that b 6 = 1 so that y = b x is not a constant function. Fact 1. The function y = b x is defined for all x and (1.1) b x > 0 for all x . Exponential functions have very useful algebraic properties : Fact 2. a. b 0 = 1 b. b 1 = b c. b x 1 + x 2 = b x 1 · b x 2 d. b - x = 1 /b x e. b ax = ( b a ) x = ( b x ) a As mentioned in Fact 1, exponential functions are defined for all x . The graph of y = b x has one of two characteristic forms, depending on whether 0 < b < 1 or 1 < b . The list includes some redundancy, but these are the algebraic properties that we use the most, so I listed them all. 1
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2 Fact 3. If b > 1 then the function y = b x increases (very rapidly) as the variable x increases, and approaches 0 when x takes very large negative values. If 0 < b < 1 then y = b x approaches 0 as x increases, and grows very large when x takes large negative values. Exponential functions grow very rapidly (when the base is greater than 1). People often use the expression ‘ growing exponentially ’ to mean growing rapidly. To get a sense of how fast exponential growth really is, read on. Fact 4. If b > 1 then the function y = b x grows (eventually) more rapidly than any power of the variable x . In other words, for any power k , no matter how large b x > x k , once x is large enough. To get a better idea of what this is saying look at a numerical example. If b = 1 . 1 and k = 100, then when x is relatively small x 100 will be bigger than 1 . 1 x , as the first few entries in the table below indicate. However when x increases, 1 . 1 x grows larger than x 100 , and the difference, 1 . 1 x - x 100 , grows exponentially with x , (remember, 10 k is a 1 followed by k zeros). x x 100 1 . 1 x 1 . 1 x - x 100 2 1 . 26 × 10 30 1 . 21 - 1 . 26 × 10 30 10 10 100 2 . 5937 - 10 100 100 10 200 13780 - 10 200 1000 10 300 2 . 47 × 10 41 - 10 300 10000 10 400 8 . 449 × 10 413 8 . 449 × 10 413 100000 10 500 1 . 855 × 10 4139 1 . 855 × 10 4139 1.1 Compound interest Exponential functions provide one of the simplest and most important models for growth, whether its growth of a population in the biological setting or growth in the value of an investment. An investment with interest rate r , compounded annually, grows according to the simple rule that at the end of each year the amount r · S is added to the investment, where S is the value of the investment at the end of the previous year. The initial amount invested, P 0 , is called the principal. We use the variable t to measure years, and let S ( t ) denote the
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