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Exponential & Logarithmic Functions Definitions, Properties & Formulas Properties of Exponents Property Definition Product x a xb x a b Quotient xa x a b , where x 0 b x Power Raised to a Power (xa)b = xab Product Raised to a Power (xy)a = xa ya Quotient Raised to a Power x xa a , where y 0 y y Zero Power x0 = 1, where x 0 a Negative Power x n 1 , where x 0 xn 1 n Rational Exponent x n x for any real number x 0 and any integer n > 1 and when x < 0 and n is odd N = N0 (1 + r)t Exponential Growth/Decay Compound Interest (Periodic) Exponential Growth/Decay (in terms of e) Continuously Compounded Interest where: N is the final amount, N0 is the initial amount, t is the number of time periods, and r is the average rate of growth(positive) or decay(negative) per time period r A P1 n nt where: A is the final amount, P is the principal investment, r is the annual interest rate, n is the number of times interest is compounded each year, and t is the number of years N = N0 ekt where: N is the final amount, N0 is the initial amount, t is the number of time periods, and k (a constant) is the exponential rate of growth(positive) or decay(negative) per time period A = Pert where: A is the final amount, P is the principal investment, r is the annual interest rate, and t is the number of years 1 Logarithmic Functions are inverses of exponential functions a logarithm is an exponent! when no base is indicated, the base is assumed to be 10 log x log10 x Common Logarithms log x y 10 y x Change of Base Formula loga n logb n logb a where a, b, and n are positive numbers, and a 1, b 1 instead of log, ln is used; these logarithms have a base of e Natural Logarithms ln x loge x ln x = y e y x all properties of logarithms also hold for natural logarithms Properties of Logarithmic Functions If b, M, and N are positive real numbers, b 1, and p and x are real numbers, then: Definition Examples logb 1 0 written exponentially: b0 = 1 logb b 1 written exponentially: b1 = b logb b x x written exponentially: bx = bx blog b x x , where x > 0 10log 10 7 7 logb MN logb M logb N log3 9x log3 9 log3 x log 1 yz log 1 y log 1 z 5 logb 2 log 4 2 log 4 5 5 7 log 8 log 8 7 log 8 x x log2 6 x x log2 6 logb M p logb M p if and only if 5 log 4 M logb M logb N N logb M logb N 5 log 5 y 4 4 log 5 y M=N log6 (3x 4) log6 (5x 2) ( 3 x 4 ) ( 5 x 2) 2 Properties of Logarithmic Functions If b, M, and N are positive real numbers, b 1, and p and x are real numbers, then: Definition Examples logb 1 0 written exponentially: b0 = 1 logb b 1 written exponentially: b1 = b logb b x x written exponentially: bx = bx blog b x x , where x > 0 10log 10 7 7 logb MN logb M logb N log3 9x log3 9 log3 x log 1 yz log 1 y log 1 z 5 logb 2 log 4 2 log 4 5 5 7 log 8 log 8 7 log 8 x x log2 6 x x log2 6 logb M p logb M p if and only if 5 log 4 M logb M logb N N logb M logb N 5 log 5 y 4 4 log 5 y M=N log6 (3x 4) log6 (5x 2) ( 3 x 4 ) ( 5 x 2) Common Errors: logb M logb M logb N logb N logb (M N) logb M logb N (log b M) p logb M p logb M logb N logb M N logb M cannot be simplified logb N logb M logb N logb MN logb (M N) cannot be simplified p logb M logb Mp (log b M)p cannot be simplified 3 College Algebra: Functions and Models Review- Exponential and Logarithmic Functions part 1 Name:____________________________________ Date:_____________________________________ ANSWER THE FOLLOWING QUESTIONS ON A SEPARATE SHEET OF PAPER AND SHOW ALL WORK! Write each expression in terms of simpler logarithmic forms: 4 5 (1) 5 log b x y (2) log b s u7 (3) logb 1 c8 (4) logb m 5n 3 p Given loga n, evaluate each logarithm to four decimal places: (5) log3 42 (6) (7) log 1 5 log6 0.00098 2 Solve each equation and round answers to four decimal places where necessary: (8) log2 x 3 (10) 1000 75e0.5 x (12) log 7 1 x 49 (9) log5 4 log5 x log5 36 (11) log 6 x 2 (13) log x 4 1 2 (14) 10 x 27.5 (15) log x log 5 log 2 log( x 3) (16) log x log 2 1 (17) log4 x 3 (18) log9 (5 x) 3 log9 2 (19) log 20 log x 1 (20) 2 1.002 4 x (21) e 25 x 1.25 (22) log( x 10) log( x 5) 2 (23) log 6 216 1 log 6 36 log 6 x 2 4 College Algebra: Functions and Models Review- Exponential and Logarithmic Functions part 2 Name:____________________________________ Date:_____________________________________ SHOW ALL WORK: (1) Anthony is an actuary working for a corporate pension fund. He needs to have $14.6 million grow to $22 million in 6 years. What interest rate (to the nearest hundredth of a percent) compounded annually does he need for this investment? (2) The number of guppies living in Logarithm Lake doubles every day. If there are four guppies initially: a. Express the number of guppies as a function of the time t. b. Use your answer from part (a) to find how many guppies are present after 1 week? c. Use your answer from part (a) to find, to the nearest day, when will there be 2,000 guppies? 5 SHOW ALL WORK: (3) The relationship between intensity, i, of light (in lumens) at a depth of x feet in Lake Erie is given by i log 0.00235 x . What is the intensity, to the nearest tenth, at a depth of 40 feet? 12 (4) Tiki went to a rock concert where the decibel level was 88. The decibel is defined by the formula i D 10 log , where D is the decibel level of sound, i is the intensity of the sound, and i0 = 10 -12 watt per i0 square meter is a standardized sound level. Use this information and formula to find the intensity of the sound at the concert. 6 SHOW ALL WORK: (5) How many years, to the nearest year, will it take the world population to double if it grows continuously at an annual rate of 2%. (6) Bank A pays 8.5% interest compounded annually and Bank B pays 8% interest compounded quarterly. If you invest $500 over a period of 5 years, what is the difference in the amounts of interest paid by the two banks? (7) Determine how much time, to the nearest year, is required for an investment to double in value if interest is earned at the rate of 5.75% compounded quarterly. 7