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Worksheet 61 (11.1) Chapter 11 Exponential and Logarithmic Functions 11.1 Exponents and Exponential Functions The rules of exponents from chapter 5 can be extended to any real number exponent. If b > 0, b 1, and m and n are real numbers, then bn = bm if and only if n = m. Summary 1: Warm-up 1. Solve: 1 81 -2x ( 3 =3 - 2x = a) 3- 2x = ) The solution set is { x= b) }. 2 4 = 64 2 2x22 ( ) = 2( ) 2x x +1 2x ( ) = 26 2 2 2x + =6 x= The solution set is { 262 }. Worksheet 61 (11.1) Problems - Solve: 1. 21 2x - 1 = 321 2. 5 2x - 6 = 119 If b > 0 and b 1, then the function f defined by f(x) = bx, where x is any real number, is called the exponential function with base b. Summary 2: Warm-up 2. Graph: a) f(x) = 3x x f(x) -2 1/9 Note: This is an example of an increasing function. -1 0 1 2 f(x) 263 x Worksheet 61 (11.1) x 1 b) f(x) = Note: This is an example of a decreasing function. 3 x f(x) -2 9 -1 0 1 2 f(x) x c) f(x) = 3x + 2 x f(x) -2 1 Note: This is a horizontal translation of f(x) = 3x. -1 0 1 264 2 f(x) x Worksheet 61 (11.1) x 1 d) f(x) = + 2 3 Note:This is a vertical translation of f (x)= 31 x f(x) -2 11 x . -1 0 1 2 f(x) x Problems - Graph: 2 3. f (x)= 5 x 265 4. f ( x ) = 2 x - 3 Worksheet 62 (11.2) 11.2 Applications of Exponential Functions Compound interest is an example of an exponential function. General formula for compound interest: nt r A= P 1+ ; n where P = principal, n = number of times being compounded, t = number of years, r = rate of percent, A = total amount of money accumulated. Summary 1: Warm-up 1. a) Find the total amount of money accumulated for $2000 invested at 12% compounded quarterly for 5 years. r A= P 1+ n nt ( A = 2000 1 + ( ( A = 2000 ( ) ) ) ( )( ) ) A= The total amount of accumulation is __________. Problem 1. Find the total amount of money accumulated for $1500 invested at 10% compounded monthly for 3 years. 266 Worksheet 62 (11.2) Summary 2: n 1 As n gets infinitely large, the expression 1 + approaches the number n e, where e equals 2.71828 to five decimal places. The function defined by f(x) = ex is the natural exponential function. Note: Use the ex key on the calculator to find functional values for x. Formulas involving e: 1. A = P ert Used for compounding continuously. A = total accumulated value, P = principal, t = years, r = rate 2. Q(t) = Q0 ekt Used for growth-and-decay applications. Q(t) = quantity of substance at any time, Q0 = initial quantity of substance, k = constant, t = time 267 Warm-up 2. a) The number of bacteria present in a certain culture after t hours is given by the equation Q = Q0 e0.3t, where Q0 represents the number of bacteria initially. If 18,149 bacteria are present after 6 hours, find how many bacteria were present in the culture initially. Q = Q0 e k t = Q0 e( )( ) Q0 = There were __________ bacteria initially. Problem 2. Find the total accumulated money for $2000 invested at 12% compounded continuously for 5 years. Worksheet 63 (11.3) 11.3 Logarithms If r is any positive real number, then the unique exponent t such that t = r is called the logarithm of r with base b and is denoted by log b r. Summary 1: 268 log b r = t is equivalent to bt = r. For b > 0 and b 1, and r > 0, 1. log b b = 1 since b1 = b. 2. log b 1 = 0 since b0 = 1. 3. r = b log b r since log b r = t. Warm-up 1. Find the equivalent exponential expression: a) log 5 125 = 3 is equivalent to 5 ( ) = 125. b) log 10 10000 = 4 is equivalent to 10 ( c) log 2 32 = 5 is equivalent to 2 ( ) ) = 10000. = 32. Warm-up 2. Find the equivalent logarithmic expression: a) 10-3 = 0.001 is equivalent to log 10 0.001 = _____. b) 1 3 4 = 811 is equivalent to log 1 3 1 81 = . c) 54 = 625 is equivalent to log 5 625 = _____. Problems 1. Find the exponential expressions for log 3 27 = 3 and log 10 .00001 = -5. 269 2. Find the logarithmic expressions for 102 = 100 and 41 3 = 641 . Worksheet 63 (11.3) Warm-up 3. a) Evaluate log 3 243 by first rewriting in exponential form and then solving. (See summary 1 in section 10.1.) Let log 3 243 = x This is equivalent to: 3x = 243 3x = 3 ( ) x = _____ Therefore, log 3 243 = _____. log 32 x = 52 b) Solve: ( 32 5 32 ) =x ( ) =x _____ = x The solution set is { }. Problems 3. Evaluate log 10 10000 by first rewriting in exponential form and then solving. 4. Solve: log 125 x = 23 For positive real numbers b, r, and s where b 1, log b rs = log b r + log b s Summary 2: 270 Warm-up 4. a) If log 10 2 = 0.3010 and log 10 7 = 0.8451, evaluate log 10 14. log 10 14 = log 10 (2 _____ ) = log 10 2 + _______________ = __________ Worksheet 63 (11.3) b) If log 2 7 = 2.8074 and log 2 5 = 2.3222, evaluate log 2 35. log 2 35 = log 2 ( _____ _____ ) = ____________ + ____________ = __________ Problems 5. If log 10 3 = 0.4771 and log 10 5 = 0.6990, evaluate log 10 15. 6. If log 2 7 = 2.8074 and log 2 3 = 1.5850, evaluate log 2 21. Summary 3: r Forlog positive real lognumbers b s = b r - logb, b sr, and s where b 1, Warm-up 5. a) If log 10 101 = 2.0043 and log 10 23 = 1.3617, evaluate . log 10 101 23 = log 10 101 - __________ log 10 101 23 = __________ b) If log 8 5 = 0.7740, evaluate log 8 645 . (Recall: 82 = 64) log 8 645 = log 8 64 - __________ = ______ - .7740 = __________ Problems 7. If log 10 3 = 0.4771 and log 10 5 = 0.6990, evaluate log 10 53 . Worksheet 63 (11.3) 271 8. If log 2 5 = 2.3219, evaluate log 2 5 32 . For positive real numbers b, r, and p where b 1, log b r p = p (log b r) Summary 4: 1 Warm-up 6. a) If log 10 1995 = 3.2999, evaluate log 10 (1995 )2 . 1 log 10 (1995 )2 = ( ) log 10 1995 = __________ b) Express as a simpler logarithmic expression: log b 3 log b 3 x y ( = log b xy x y ) ) log b xy =( = 31 ( - ) c) Solve: log 3 (2x - 1) + log 3 (x + 1) = 2 log 3 ( )( )=2 32 = ( )( ) 9 = _______________ 0 = _______________ x = _____ or x = _____ Note: Logarithms are only defined for positive numbers. Negative results are extraneous. The solution set is { Worksheet 63 (10.3) 272 }. Problems 9. If log 10 5 = 0.6990, evaluate log 10 54. 10. Express as a simpler logarithmic expression: log b x2 y 11. Solve: log 5 (4x + 1) - log 5 (x - 1) = 1 Worksheet 64 (11.4) 11.4 Logarithmic Functions 273 A function defined by an equation of the form f(x) = log b x, b > 0 and b 1 is called a logarithmic function. y = log b x is equivalent to x = by. f(x) = bx and g(x) = log b x are inverse functions. Summary 1: Warm-up 1. a) Graph: y = log 3 x Note: This is the inverse of y = 3x from warm-up 2a in section 10.1. Inverses are reflections of each other through the line y = x. b) Graph: f(x) = log 3 (x - 2) Note: This is a horizontal translation 2 units right. 274 Worksheet 64 (11.4) Problems Note: See warm-up 2b in section 10.1. 1. Graph: f(x) = log 1 x 3 2. Graph: f(x) = 2 + log 1 x 3 Logarithms with a base of 10 are called common logarithms. Summary 2: 275 log 10 x = log x Note: Use log key on calculator to evaluate common logarithms. f(x) = log x and g(x) = 10x are inverse functions. Warm-up 2. Evaluate to four decimal places: a) log 1.25 = __________ b) log 12.5 = __________ c) log 125 = __________ d) log 1250 = __________ Worksheet 64 (11.4) Problems - Evaluate to four decimal places: 3. log 0.0243 4. log 0.243 5. log 2.43 6. log 24.3 Warm-up 3. Find x to five significant digits: a) log x = 0.4150 Note: Use 10x key on calculator to find x. ) x = 10( x = __________ b) log x = 1.6135 276 ) x = 10( x = __________ Problems - Find x to five significant digits: 7. log x = 0.0101 8. log x = -4.321 Natural logarithms are logarithms that have a base of e. log e x = ln x Note: Use ln key on calculator to evaluate natural logarithms. f(x) = ln x and g(x) = ex are inverse functions. Summary 3: Warm-up 4. Evaluate to four decimal places: a) ln 1.25 = __________ b) ln 12.5 = __________ c) ln 125 = __________ d) ln 1250 = __________ Worksheet 64 (11.4) Problems - Evaluate to four decimal places: 9. ln 0.0243 10. ln 0.243 11. ln 2.43 12. ln 24.3 Warm-up 5. Find x to five significant digits: 277 a) ln x = 0.4150 Note: Use ex key on calculator to find x. ) x = e( x = __________ b) ln x = 1.6135 ) x = e( x = __________ Problems - Find x to five significant digits: 13. ln x = 0.0101 14. ln x = -4.321 Worksheet 65 (10.5) 11.5 Exponential Equations, Logarithmic Equations, and Problem Solving Summary 1: 278 If x > 0, y > 0, and b 1, then x = y if and only if log b x = log b y. Warm-up 1. Solve to the nearest hundredth: a) b) 10x = 5 log ( ) = log ( ) ( )log 10 = log 5 ( ) x= ( ) x = __________ The solution set is { }. The solution set is { }. 5x + 1 = 7 log ( ) = log ( ) ( )log 5 = log ( ) ( ) x+1= ( ) x = __________ c) ln (x + 2) = ln (x - 3) + ln 2 ln (x + 2) = ln [2( )] x + 2 = ____________ x = _____ The solution set is { Problems - Solve to the nearest hundredth: 1. ex + 1 = 40 2. 72x = 11 Worksheet 65 (11.5) 3. log (x - 1) + log (x - 4) = 1 279 }. Warm-up 2. Use the compound interest formula A = P 1 + nr logarithms to solve: n t and a) How long will it take $1000 to double itself if invested at 10% interest compounded quarterly? (Round to tenths.) A = P 1 + nr ( n t ( ) )= ( ) 1 + ( ) 4t 2=( ) log 2 = log 1.025 t = _________ ( )t It will take _____ years. Problem - Use the formula A = Pert and natural logarithms to solve: 4. How long will it take $1000 to double itself at 10% interest when compounded continuously? (Round to nearest tenth.) Worksheet 65 (11.5) The change-of-base formula for logarithms: Summary 2: 280 log a r = log b r ; where a, b, and r are positive real numbers log b a and a 1 and b 1. Warm-up 3. Approximate to 3 decimal places: a) log 3 15 log ( ) log ( ) __________ log 3 15 = Note: Either common or natural logarithms can be used to approximate logarithms with bases other than 10 or e. b) log 5 0.004 log ( ) log ( ) __________ log 5 0.004 = Problems - Approximate to 3 decimal places: 5. log 6 88 6. log 2 0.001 281