Math Logarithms and Exponentials Review Key

Algebra 2 Midterm Review Logarithms & Exponentials Name Hour Let f(x) = x²+3x-1 and g(x)=4x-x². Perform the indicated operation and state the domain when necessary. 2. f(g(2)) g(2)=8-4 =4 1. g(-2) =4(-2)-(-2)2 --8-4 -12 4. g(a+2) = 4 (a+2)-(a+2)² =4a+8-(a²+Ya+4) =4a+8-a²-4a-4 = -a²+4 Find the equation for f(x). 7. f(x) = 2x-5 E f(4)=(4)²+3(4)-1 =16+12-1 = 27 5. f(b-1) =(b-1)² +316-1)-1 b26+1+36-3-1 = b²+b-3 3. g(f(x)) 4(x²+3x-1)-(x²+3x-1) =4x²+12x-4-(x²+bx²+7x²-6x =-x²-6x³-3x²+18x-4 6. f(x+2) = (x+2)²+3(x+2)-1 = x²+ 4x +4 +3x+6-1 = x²+7x +9 y = 2x-5 3x +5 = 2y 3 3x+5 X=2y-5 3x = 2y-5 y f(x)= 3x+5 8. f(x)=-2 T =-2 X+2 f(x)=1 9. (x)=√x+1 y=√x+1 f/x)=x²-1 Trade X=vy+1 2 x²-1=y Without Graphing, determine whether or not the following functions have inverse functions. X+2= y y(x+2)=1 x+2 x²=y+1 10. f(x) = x²+17 Fails Horizontal Line Test-no 11. f(x)=2x-18 + п (yes 12. f(x)=5x+17x-8 Fails Horizontel Line Test-no Write in Log form. 13. 2532 Write in Exponential Form. 14, 100.1 15. In 3 = x 16. logx = 7 2 1092 32:5 log 0.1 = -1 ex=3 7/2 10 =X Solve. 1 20. log16x = 162=x 4=x 17. logx = 2 18. loggx = 19. loga x = 2.5 82.5=X 2√2x 8512 = x 21512X 2 6² = x 36 = x 93/2 = x 33 = x X 27 = X Condense the following logarithmic expressions into a single logarithm. 21. 2 log P+ log Q 109 P² 109 Q b b b 23. 4-2 logs a log, 6-log, a² 1096(6) Solve the following logarithmic equations. 25. logax 2 loga 3 + loga 5 logax = loga 3.5 logax = 109 45 X=45 26. log(x+1)+log x = 1 log, x(x+1)=1x33x=2 X 6' = x(x+1) 0=(x+3)(x-2) 519601 = (39) 9601 22. logs 6+ loga 5-loga 2 24. 2log3m-logan-3-log32 3-log3 2 log3 m²-log, Vn-1093 3 3 m² 1093.27.2 1093 3 54 = 28. logx-log(x-1)= x=2(x-1) 27. Inx+ln(x+3) = In 10 In x(x+3) = In 10 x²+3x = 10 2 X2+3x-10-0 (x+5)(x-2)=D 1094 X = logy 42 1-x X=2x-2 1094 X-1 10942 -X=-2 X X-1 2. 31. 32993x-1 X=2 Solve the following exponential equations. 29. 25125 52x53 2x =3 x = 3/2 30.4 22x = 2-x+3 2x-x+3 3x3, X x+1=6x-2 13/25x 3x+ = 3 6x-2 13 = x 32. 25 In 21-x = In 5 (1-x) In 2 In 5 1-x = 145 142 -17511-1 x= 5 +1 =-1.322 33. 7x+1=5x+9 In 7x+1= In 5x+9 (x+1)7= (x+4)5 xlm7+17= xln5+9 145 Xin7-xin59m5-17 x=91n5-1n737.26 34. 12+35x20 358 In 3 = In 8 sx In3 = In8 x: Ing 20.379 5in 3 In Solve the following exponential equations. 35, 2-3x12 36. 4e2x = 7 In e 2-3x In 12 2-3x= In 12 Inezx= In 1 -3x= In 12-2 2x = In In 12-2 -3 2 -0.162 0.280 Let u= ex 37. ex+5e2x-6=0 4²+54-6:0 (u+b) (4-1)=0 4-6 u=1 e2x e2x=1 2x=bul 38. Of an original 100 g of a radioactive substance, 70 g remained after 12 days. How much will N=100( N=No(½)th be left after 20 days? 0.7=(는) In 0.7 = In (½) m(t) 12 h 70=100(±)/ 1n0.7=1m(s) 12 17 h 23.320 10.3 h.no.7:12 In ~55.186 39. The population of Ubetchakan grew from 65.6 million in 1975 to 80 million in 1985. What is the rate of annual growth? A = P(1+r)t 80=65.6(1+r) 1.220: (1+r)" 1.020081+r 0.02008 = r 2% ≈ r 40. The value of a new $3500 sailboat decreases 10% per year. Find its value after 20 years. A = P(1-r)t A= 3500 (.90) 20 A≈ 1425.52 41. You are saving for your senior year spring break trip. You want to have $3000 in your savings account after 3 years. Find the amount you should deposit for each of the situations described. a. The account pays 2.25% annual interest compounded quarterly. 3000 = P(1+:02.25.) 4.3 3000=P(1.06963) $2804.71= P b. The account pays 3.5% annual interest compounded monthly. 3000 = P(1+03) 12.3 3000 = Pe (04) (3) c. The account pays 4% annual interest compounded continuously. P=42,701.39 P= $.2,660.76 42. A man invests $10,000 in an account that pays 6.5% interest per year, compounded quarterly beginning on the first of February of 2006. a. Find the amount after 6 years. A= 10,000 (1+065) 4.6 = $14,723.58 = b. During which month of what year will the amount triple? 30,000=10,000 (1+005) 4t 3=(1.01625) In 3 = 4t ln (1.01625) 17.038 February of 2023 43. During which month of what year will it take for an investment of $2,000 deposited on February 1", 1985 to double in value if the interest rate is 10.5% per year, compounded continuously? 4,000= 2000 e .105t 2 = e.lost .105t In 2 .105t Sept of 1991 n2t~6.601 yrs .105 byrs + 7.2 months. 44. A culture contains 1,600 bacteria initially and doubles every 30 minutes. a. Find a function that models the number of bacteria at time t. N=1600 (2) +130 b. Find the number of bacteria after three hours. 180 minutes N=1600 (2) 180/20 = 102,400 c. After how many minutes will there be 6,000 bacteria? 6000=1600 (2) 130 3.75=2+/30 In 3.75 /30. In 2 +57.2 minutes In 3.75= In 24/30 In 3.75 1n2 t 30 = Radium-123 has a half-life of 1,400 years. Suppose a sample of this substance has a mass of 32 mg. a. Find a function that models the amount of the sample remaining at time t N=32.()+/1400 b. Find the mass remaining after 6,000 years. N 32(½) 6000/ 1400~1.641 mg c. How long will it take for the sample to decay to a mass of 12 mg? 12=32(2)+livor 0.375: ()+/1400 In 0.375 = In 0.375 In In 112 1400 +~1,981.052 yrs. Without using a calculator, graph the exponential equation f(x)=25. Then, graph each of the transformed functions. Be sure to list all transformations in the order in which they must be graphed. Then, find the Domain and Range. 46. g(x)=-2-2 reflect over x-axis vertical stretch 47. h(x)=2x+3-3 left 3, down 3 Domain: (-50, 60) Range: (0, 0) Domain: (-to, bo) Range: (-3,00) y = log3x means Without using a calculator, graph the logarithmic function f(x) = logs x. Then, graph each of the transformed functions. Be sure to list all transformations in the order in which they must be graphed. Then, find the Domain and Range. 48. f(x)=log3(x+4) left 4, reflect over X-axis Domain: (-4,00) Range: (-Do, 00) 49. f(x)= 2 log3x-4 Vertical stretch, down 4 Domain: (0,00) Range: (-00, DO) X 9-2 13 1 3 १ 2 1