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Unformatted text preview: Growth Authors: Bill Davis, Horacio Porta and Jerry Uhl ©1996-2007 Publisher: Math Everywhere, Inc. Version 6.0 1.02 Natural Logs and Exponentials BASICS B . 1) The natural base e and the natural logarithm After , 1 , and p , the most important number in calculus is e . Some folks would argue this point, saying that e is the most important number in calculus. As you progress through the course, you can decide for yourself. Here is e to 10 accurate digits: N @ E, 10 D 2.718281828 Lots of folks like to say that e = 2.7 Andrew Jackson Andrew Jackson because Andrew Jackson was elected President of the United States in the year 1828. Mathematica can slam out the decimals of e . Here is e to 50 accurate digits ( 49 accurate decimals): N @ E, 50 D 2.7182818284590452353602874713526624977572470937000 One hundred accurate digits: N @ E, 100 D 2.71828182845904523536028747135266249775724709369995957496696762772 Ö 4076630353547594571382178525166427 No one has ever succeeded in finding a pattern for the decimals of this number. In this sense, this number is squirrelly as all get out. But for reasons that you will see after just a few more lessons, e is the natural base for exponentials and logarithms. · B.1.a) Plot e x and then plot e- x and describe what you see. · Answer: Here is a plot of e x : Clear @ x D ; Plot @ E x , 8 x, 0, 7 < , PlotStyle-> 88 Blue, Thickness @ 0.015 D<< , AspectRatio-> 1 ê GoldenRatio, AxesLabel-> 8 "x" , "E x " <D 1 2 3 4 5 6 7 x 100 200 300 400 500 600 E x Pristine exponential growth. Here is a plot of e- x : Clear @ x D ; Plot @ E- x , 8 x, 0, 7 < , PlotStyle-> 88 Blue, Thickness @ 0.015 D<< , PlotRange-> All, AxesLabel-> 8 "x" , "E- x " <D 1 2 3 4 5 6 7 x 0.2 0.4 0.6 0.8 1.0 E- x Pristine exponential decay. · B.1.b.i) The natural logarithm The natural logarithm is the logarithm function whose base is e . Every scientific calculator has a button that activates this function; so you know it's a pretty big deal in science. Saying that y = Log @ x D is the same as saying e y = x . Try it out: x = 12; y = N @ Log @ x DD 2.48491 E y 12. x == E y True Play with other x 's and rerun. Plot Log @ x D and describe what you see. · Answer: Clear @ x D ; Plot @ Log @ x D , 8 x, 0, 8 < , PlotStyle Ø 88 Blue, Thickness @ 0.015 D<< , AxesLabel Ø 8 "x" , "Log @ x D " <D 2 4 6 8 x- 1 1 2 Log @ x D Logarithmic growth is fast at first and then very slow later, just the opposite of exponential growth. · B.1.b.ii) The main laws of exponents are e a + b = e a e b e a- b = e a e b H e a L b = e a b . You've seen these before and in previous math courses you saw how these laws of exponents are related to the main properties of Log @ x D : Log @ a b D = Log @ a D + Log @ b D Log A a b E = Log @ a D- Log @ b D Log A a b E = b Log @ a D ....
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