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STUDY GUIDE FOR CHAPTER 5 In this chapter Student should be able to: a) Solve exponential equations b) Graph exponential function c) Change a function from logarithmic to exponential form and from exponential to logarithmic form d) Apply lows of logarithms e) Find the logarithms using calculator f) Solve logarithmic equation g) Solve problems involving compound interest h) Solve problems involving exponential growth and decay. 5.1. INVERSE FUNCTION Definition of one-to-one function: If a b are elements of a domain of f(x) and if f(a) f(b) then f(x) is one-to-one function. A function in which elements from the domain always lead to different elements from the range is called on-to-one function. Examples: Is f(x) one-to-one function? f(x) = 3x + 2 f ( x) x 2 4 For every x there is only one y For x = 2 and for x = - 2 there is only one y = 0 Horizontal Line test for one-to-one function: Intersects with f(x) in no more then one point if f(x) is one-to-one function’ Graph shows that parabola is not a one-to-one function. Inverse Function: If f(x) and g(x) are inverse function then: f[g(x)] = x and g[f(x)] = x If f(x) is given how to find g(x) = f -1(x): Procedure for finding one-to-one function: 1. 2. 3. 4. Check that f(x) is one-to-one function. Solve for x [x = f -1(y)]. Exchange x and y [y = f -1(x) = g(x)]. Check that f[g(x)] = x and g[f(x)] = x The domain of f(x) equals the range of f -1(x) and the range of f(x) equals the domain of f -1(x). Practice: Find the inverse of: 1. y = f(x) = 4x – 5 y5 4 x5 y f 1 ( x ) g ( x ) 4 x a) Solve for x: b) Exchange x and y: c) Check that f[g(x)] = x f [ g ( x) 4 x5 5 x 4 and g[ f ( x ) 4x 5 5 x 4 y= x y=(x + 5)/4 y=4x - 5 2. y = 2x + 1 3. y 6 x 4. y = - x2 + 2 5. y = 1/x and g[f(x)] = x 5.2. EXPONENTIAL FUNCTIONS Properties of Exponents: If x = y then ax = ay If a > 1 and x < y then ax < ay If 0 < a < 1 and x < y then ax > ay Graphing exponential function f(x) = ax a) The points (0,1) and (1,a) are in the graph b) If a > 1 f(x) is increasing c) If 0 < a < 1, f(x) is decreasing d) x - axis is horizontal asymptote e) Domain (No restriction), Range ( y > 0) Graph the following functions: f(x) = 2x 5 y y =2x 4 3 2 ( 1 ,2 ) 1 x 0 -5 -4 -3 -2 -1 -1 0 1 2 -2 -3 -4 -5 f(x) = 3 x f(x) = - 3 x f(x) = 3x+2 f(x) = 3 x + 2 f(x) = 3x+ 2 + 2 f(x) = 2 x + 1 f(x) = (1/2)x f(x) = 2 –x f(x) = - 2 x 3 4 5 Solve exponential equations using the property: if a x = a y then x = y: (1/3) x = 81 Solution: 1 ( ) x 36 3 1 1 ( ) x ( ) 6 3 3 x 6 (3-1) x = 34 (2/3)x = 9/4 23-x = 8 16 x + 2 = 8 1 - x 32t = 16 1 - t (2/3)k - 1 = (81/16) k + 1 81 = x 4/3 5 2p + 1 = 25 Applications of Exponential Equations Q 48 on page 368 Q 50 on page 368 Q 52 on page 368 Compound Interest If P dollars are deposited in an account paying an annual rate of interest r, compounded m times per year, then after t years the account will contain A dollars, where A P(1 r mt ) m A – future value P – present value Example 1: How much should be deposited at 6 % interest compounded annually (m = 1) for three years ( t = 3) in order to have a balance of $20,000 in three years? 0.06 3 ) 1 20,000 P(1.191016) 20,000 P(1 P 16,792.39 Example 2: If only $15,000 is deposited , what annual interest rate is required to increase to $20,000[10 %] Example 3: Invest $1 (P = 1) for one year ( t = 1) at the rate of 100 % (r = 1) and find A compounded 1, 2, 5, 10, 25, 100, 500, 1000, 10000, 100000, 1000000..... times a year. Future value should go toward fixed number e = 2.71828 Compounded annually: 1 P 1 (1 )1 2 1 Compounded twice a year: 1 P 1 (1 ) 2 2.25 2 Compounded ten times a year P 1 (1 1 10 ) 2.5937 10 Compounded one hundred times a year P 1 (1 1 100 ) 2.7048 100 Compounded one thousand times a year P 1 (1 1 1000 ) 2.7169 1000 Compounded one hundred thousand times a year P 1 (1 1 )100,000 2.71826 100,000 Compounded one million times a year P 1 (1 1 )1,000,000 2.71828 1,000,000 Compounded continuously P 2.71828 e Exponential Growth and Decay y = yo e kt y – future amount or number yo – is amount or number at present time t = 0 k - constant Work on Q. 56 on page 369 5.3. LOGARITMIC FUNCTION How to find the inverse of a exponential function y = ax – Graph. 1. It’s one-to-one function 2. How to solve for x ? Introduction of Logarithmic functions y = ax Solving for x: f(x) = a x x = log a y f –1(x) = log a x Exchange x and y: y = log a x Meaning of a logarithm: Exponential form 25 = 5x Logarithmic form: x = log 5 25 “Five raised to what exponent is 25” Log function is an Inverse of exponential function Write eq. in logarithmic or exponential form: a) b) c) d) e) f) x=ay (1/2)-4 = 16 51=5 (3/4)0 = 1 9 2.7 = z 0.315=2 –5 y = loga x -4 = log ½ 16 1 = log 5 5 0 = log ¾ 1 2.7 = log 9 z -5 = log 2 0.3125 Properties of Logarithms log a xy = log a x + log a y log a x/y = log a x - log a y log a x r = r log a x log a a = 1 log a 1 = 0 Application of properties log a (m n q/p2) = ? =log a ( m n q ) – log a p 2 = = log a m + log a n + log a q – 2 log a p log6 3 m2 ? x y log a ? z3 x3 y5 ? zm logb log a x + log a y – log a m = ? 2 log m a – 3 log m (b2 ) = ? (log b k – log b m ) – log b a = n log b k k log b a log b m ma Use graphing calculator to graph logarithmic functions: f(x) = log 10 ( x – 1) f(x) = log 10 x - 1 5.4 EVALUATING LOGARITHMS Common logarithm: log x = log 10 x Natural number e = 2.73… Natural logarithm: log e x = ln x Use calculator to evaluate ( to the nearest ten-thousands ) log 1247 = ? log0.0069 = ? ln0.7 = ? ln580 = ? ln650 = ? log312.5 = ? log 10 = ? ln 1 = ? ln0.049 = ? ln4.44 = ? Change of base theorem: log b x log x ln x log b a log a ln a log a x log 9 12 = log 12/log 9 = ? log2.9 7.5 = log 7.5/log 2.9 = ? log 5 17 = log 2 0.1 = 5.5 EXPONENTIAL AND LOGARITHMIC EQUATIONS Property: When x > 0, y > 0, a > 0 ( a 1 ) and if log a x = log a y then x = y Solve exponential equations: 5 x = 14 ln( 5 x ) ln 14 x ln( 5) ln( 14) x ln 14 1.64 ln 5 3 4x – 1 = 4 x – 2 3 2x – 5 = 13 e 2 – y = 12 e 2x e 5x = e 14 Solve logarithmic equations: ln (5 + 4y ) – ln (3 + y) = ln 3 5 4y ln 3 3 y 5 4y 3 3 y 5 4 y 3(3 y ) ln 4 y 3y 9 5 y4 ln x – ln (x + 1) = ln 5 ln ( k - 5) + ln ( k+ 2) = ln (14 k) ln e x – ln e 3 = ln e 5 5.6 EXPONENTIAL GROWTH AND DECAY In many applications: Biology Economics Social studies Physics a quantity changes at a rate proportional to the amount present At any time (instant) t, the amount present is: y y o e kt y is future amount yo is present amount (at t = 0) k – constant (k > 0 growth, k < 0 decay) Example: A sample of 500 g of plutonium 341, decays according to the function: A(t ) Ao e 0.053t where t is time in years. A(t) is future amount Ao id present amount Find the amount of the sample remaining after: a) 4 years b) 8 years c) 20 years d) Find the half life Solution: a) b) c) t4 A(4) 500 e 0.0534 500 e 0.212 404 g t 8 A(8) 500 e 0.0538 327 g t 20 A(20) 500 e 0.05320 173g The HALF-LIFE of quantity that decays exponentially is the time it takes for half of a given amount to decay. 500 250 2 250 500 e 0.053t A(t ) d) 1 e 0.053t 2 1 ln( ) 0.053 t 2 1 ln( ) 2 13 years t 0.053 Example: By Newton’s law of cooling, the temperature of a body at time t after being introduced into environment having constant temperature To is: T (t ) To C e kt where C and k are constants. If C = 100 and k = 0.1, and t is time measured in minutes, how long will it take a hot cup of coffee to cool to a temperature of 25 oC in a room at 20 oC. A(t ) 25o C To 20o C 25 20 100 e 0.1t 5 100 e 0.1t 1 e 0.1t 20 1 ln( ) 0.1t 20 t 30 min Example: How long it take for $5000 to grow to $8400 at an interest rate of 6 % if interest is compounded a) semiannually and b) continuously. a) r A P (1 ) mt m 0.06 2t 8400 5000(1 ) 2 84 1.025 e 2 t 50 84 ln( ) 2 t ln( 1.025) 50 t 8.8 years b) A P e rt 8400 5000 e 0.06t 84 e 0.06t 50 84 ln( ) 0.06 t 50 t 8.64 years