Notes 3X (Logs and exponents)

Notes 3X (Logs and exponents) - MIT OpenCourseWare...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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X. EXPONENTIALS AND LOGARITHMS 1. The Exponential and Logarithm Functions. We have so far worked with the algebraic functions - those involving polynomials and root extractions - and with the trigonometric functions. We now have to add to our list the exponential and logarithm functions, since these are used in your science and engineering courses from the beginning. Your book will handle the calculus of these functions; here we want to review briefly their algebraic properties, and look at applications; one or two of them might be new to you. Where does one encounter exponentials and logarithms? In general, exponentials are used to express all sorts of simple growth and decay processes. 1. The growth of a bacteria colony which doubles in size every day: y = yo2 t , where t = time in days, y = population size, yo = the initial size, i.e., size at t = 0. 2. Dollars in a bank account, at 5%.interest compounded annually: A = Ao(l.05) n , where n = number of years, A = amount, Ao = initial amount (the "principal"). 3. Amount of radioactive substance, with a 1 year half-life: x = zo(1/2)n = zo2 - n, where n = number of years, x = amount, 0o = initial amount. There are many other examples: the decay of electric charge on a capacitor, the way a hot and cold body come to the same temperature when they are brought together, the falling of a body through a resisting medium (a steel ball dropped into oil, for instance) - all involve exponentials when you express them in mathematical terms. We use logarithms when the base of the exponential is unimportant and we want to focus our attention on the exponents instead. Suppose the base is 10; writing simply "log" for "log 0 o" in what follows, we have y = 10' logy = z. 1. Star magnitude. The observed brightness B of a star is described by comparing it to a standard brightness Bo, using the equation B = BolO - 5 . 10 The important thing is the number m called the magnitude; it is defined by the above equation, or equivalently, taking logs, by 5 B m = - log .
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