Chapter Review 375 ( ) = < < f x x a log , 0 1 a ( ) = = y x x a log if and only if a y Domain: the interval ( )∞ 0, Range: the interval ( ) −∞ ∞, x-intercept: 1; y-intercept: none Vertical asymptote: = x 0 (y-axis) Decreasing; one-to-one; smooth; continuous See Figure 44(b) for a typical graph. Properties of logarithms (pp. 327–328, 331) = = = = = a a M a r a e log 1 0 log 1 log a a M a r r r a log ln a ( ) ( ) = + = − MN M N M N M N log log log log log log a a a a a a = M r M log log a r a If = M N, then = M N log log . a a If = M N log log , a a then = M N. Formulas Change-of-Base Formula (p. 332) = M M a log log log a b b ≠ ≠ a b 1, 1, and M are positive real numbers Compound Interest Formula (p. 346) ( ) = ⋅ + A P r n 1 nt Continuous compounding (p. 348) = A Pert Effective rate of interest (p. 349) Compounding n times per year: ( ) = + − r r n 1 1 E n Continuous compounding: = − r e 1 E r Present Value Formulas (p. 350) ( ) = ⋅ + = − − P A r n P Ae 1 or nt rt Uninhibited growth or decay (pp. 355, 357) ( ) = A t A ekt 0 ≠ k 0; growth, > k 0; decay, < k 0 Newton’s Law of Cooling (p. 359) ( ) ( ) = + − < u t T u T e k 0 kt 0 Logistic model (p. 360) ( ) = + − P t c ae 1 bt > > ≠ a c b 0, 0, 0 Objectives Section You should be able to . . . Example(s) Review Exercises 5.1 1 Form a composite function (p. 273) 1, 2, 4, 5 1–6 2 Find the domain of a composite function (p. 274) 2–4 4–6 5.2 1 Determine whether a function is one-to-one (p. 281) 1, 2 7(a), 8 2 Determine the inverse of a function defined by a mapping or a set of ordered pairs (p. 283) 3, 4 7(b) 3 Obtain the graph of the inverse function from the graph of a one-to-one function (p. 285) 5 8 4 Verify that a function defined by an equation is an inverse function (p. 286) 6, 7 9, 10 5 Find the inverse of a function defined by an equation (p. 287) 8, 9, 10 11–14 5.3 1 Evaluate exponential functions (p. 294) 1 15(a), (c), 48(a) 2 Graph exponential functions (p. 298) 3–6 32–34, (a)–(c), 35(d)–(f) 3 Define the number e (p. 302) p. 301 4 Solve exponential equations (p. 304) 7, 8 36, 37, 40, 42, 48(b) (continued)
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