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For example, if we look at exponential function whose base 2, nf(64) = 2 64 =18, 446, 744, 073, 709, 525, 000 f(64) = 2 64 =18, 446, 744, 073, 709, 525, 000 And 2 n t even f(64) a very= big 2 64 number =18, 446, to744, be 073, using709, for525, a base 000 (any positive number And 2 n t can be even a base, a very bigplenty number numbers to using are much, for a base much(any bigger positive than 2). does n For example, grow if webigger look at as it moves exponential to right, function butwhose it gets base big in2, a n hurry. doesfor example, grow if we bigger lookas atit moves exponential to right, function but whose it gets base big Not in aonly 2, hurry. Not only does grow bigger as it moves to right, but it gets bignot in a hurry. becomes becomes shorter shorter shorter, shorter, shrinking shrinking towards, towards, but but never never touching, touching, x-ax. The y-intercept at 1 when moving to right, grows 180Ĥ taller taller when moving to left, becomes shorter shorter, taller taller shrinking taller taller towards, when when moving moving but never to to touching, left, left, x-ax. f a greater than 1, n =a x grows taller as it moves to right. There are two options: eir base greater than 1, or base less than 1 (but still positive). * * * * * * * * * * * * * Graphs exponential functions t s really important you know general shape an exponential function. Forthreason,weusuallydon ttalkmuch about exponential function whose base equals 1. Base 1 f anexponentialfunctionwhosebaseequals1 if =1 x nforn, m 2 N we have n f =1 m p n m m = 1 n = mp 1=1 n fact, for any real number x, 1 x =1,s(x) =1 x same function as constant function =1. For example, 3 4 =81 a 0 =1 f n, m 2 N, n a n m = m p a n =( mp a) n a x = 1 a x The rules above were designed so following most important rule exponential functions holds: 178Ģ a x a y = a x+y Anor variant important rule above a x a y = ax y And re also following slightly related rule (a x ) y = a xy Examples = 2 p 4=7 0 =15 1 =15 (2 5 ) 2 =2 10 =1024 (3 20 ) 1 10 =3 2 = 1 (8) 2 3 = 1 ( 3p 8) 2 = 1 4 * * * * * * * * * * * * * 179ģ The base an exponential function f =a x, n we call a base exponential function.
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Rules for exponential functions Here are some algebra rules for exponential functions will be explained in class. The function =3 x an exponential function variable exponent. Here variable, x, being raed to some constant power. There a big di erence between an exponential function a polynomial. For example, =3 x an exponential function, g(x) =( 4 17 )x an exponential function. For any positive number a>0, re a function f : R! (0, 1) called an exponential function defined as =a x. 1 Exponential Functions n th chapter, a will always be a positive number.