Name: Class: Date: 1. Find the exponential function f (x) = a x whose graph is given.

Transcription

1 . Find the exponential function f (x) = a x whose graph is given. 2. Find the exponential function f (x) = a x whose graph is given. 3. State the domain of the function f (x) = 5 x. PAGE

2 4. State the asymptote of the function g (x) = 9 x State the range of the function h (x) = x 6. Find the function of the form f (x) = Ca x whose graph is given. 7. If $000 is invested at an interest rate of 0% per year, compounded semiannually, find the value of the investment after 0 years. 8. The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired sum at a later date. Find the present value of $0000 if interest is paid at a rate of 9% per year, compounded semiannually, for 9 years. 9. Find value of x at which the local minimum occurs for the function f (x) = e x + e 4 x. State the answer correct to two decimal places. PAGE 2

3 0. Express the equation in exponential form log 3 3 =. Express the equation ln (x + 5) = 2 in exponential form. 2. Express the equation in logarithmic form 0 2 = Express the equation 5 z = n in logarithmic form. 4. Express the equation in logarithmic form e x + 2 = Use the definition of the logarithmic function to find x: log 6 x = 0 6. Use the definition of the logarithmic function to find x: log x 9 = 2 PAGE 3

4 7. Find the function of the form y = log x whose graph is given. a PAGE 4

5 8. Find the function of the form y = log a x whose graph is given. 9. Find the domain of the function g (x) = log 3 (x 2 4). 20. Find the domain of the function f (x) = log (x x 4 ) Find the domain of the function f (x) = x 4 log 2 x 5 PAGE 5

6 22. Use the Laws of Logarithms to rewrite the expression log 3 ( x (x 9) ) in a form with no logarithm of a product. 23. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a quotient. log 6 x Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a power. log Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a power. log 6 7 x Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient or power. log a x 8 yz Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product or power. ln 6 3r 5 s PAGE 6

7 28. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient or power. log b 5 a 9 c 29. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient or power. ln x 3 y z 30. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient or power. log x ( x 5 + ) ( x 3 2 ) 2 3. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product or power. log 5 x 5 y 5 z 32. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product or power. ln z 5 x 9 y 2 + 3y Evaluate the expression log PAGE 7

8 34. Rewrite the expression as a single logarithm log log Rewrite the expression below as a single logarithm. log 2 + log 6 log Rewrite the expression as a single logarithm ln ln x + 7 ln (x 2 + 7) 37. Simplify (log 3 5)(log 5 9). 38. Find the solution of the exponential equation, correct to four decimal places. e x = Find the solution of the exponential equation 4 4 x = 9, correct to four decimal places. 40. Find the solution of the exponential equation, correct to four decimal places x = 4 4. Find the solution of the exponential equation e 2 2 x =, correct to four decimal places. 42. Find the solution of the exponential equation 5 x = 2 x + 3, correct to four decimal places. PAGE 8

9 43. Find the solution of the exponential equation below, correct to four decimal places. 3 + e x = Find the solution of the exponential equation x = 0, correct to four decimal places. 45. Solve the equation x 7 6 x = 6 x 46. Solve the equation e 2x 5 e x + 4 = Solve the logarithmic equation for x: ln x = Solve the logarithmic equation for x: log x = 49. Solve the logarithmic equation for x: log ( 9 x + 7 ) = Solve the logarithmic equation for x: log 3 ( 4 x ) = PAGE 9

10 5. Solve the logarithmic equation for x: log log 2 x = log log 2 ( x 25 ) 52. Solve the logarithmic equation for x: log 5 ( x + 7 ) log 5 ( x 7 ) = For what value of x is the following true? log ( x + 6 ) = log x + log Solve for x. log 2 ( log 3 x ) = A sum of $0000 was invested for 7 years, and the interest was compounded semiannually. If this sum amounted to $2900 in the given time, what was the interest rate? 56. A 25 g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by m ( t ) = 25 e t where m( t ) is measured in grams. After how many days is there only 8 g remaining? PAGE 0

11 57. An electric circuit contains a battery that produces a voltage of 60 volts ( V ), a resistor with a resistance of 3 ohms ( ), and an inductor with an inductance of 5 henrys ( H ), as shown in the figure. Using calculus, it can be shown that the current I = I ( t ), ( in amps A ) t seconds after the switch is closed is I = 60 3 ( e 3t/5 ). Consider how you would express time as a function of current, and apply that to answer the following question. After how many seconds is the current 2.5 A? Enter the number of seconds rounded to three decimal places. 58. Use a graphing device to find all solutions of the equation, correct to three decimal places. log x = x Use a graphing device to find all solutions of the equation, correct to two decimal places. e x 2 2 = x 3 x 60. Solve the inequality. log ( x 6 ) + log ( 3 x ) < 6. Solve the inequality. 2 < 0 x < 7 PAGE

12 62. Solve the inequality x 2 e x 25 e x < Solve the equation 9 x 3 x + = The fox population in a certain region has a relative growth rate of 8% per year. It is estimated that the population in 998 was 000. Find a function p(t) that models the population t years after A culture starts with 8600 bacteria. After one hour the count is 000. Find a function that models the number of bacteria n ( t ) after t hours. 66. An infectious strain of bacteria increases in number at a relative growth rate of 200% per hour. When a certain critical number of bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in 28 hours. How long (in hours) will it take for the critical level to be reached if the same person is infected with 8 bacteria? 67. The half life of cesium 37 is 30 years. Suppose we have a 85 g sample. Find a function that models the mass remaining after t years. 68. Newton s Law of Cooling is used in homicide investigations to determine the time of death. The normal body temperature is 98.6 o F. Immediately following death, the body begins to cool. It has been determined experimentally that the constant in Newton s Law of Cooling is approximately k = 0.947, assuming time is measured in hours. Suppose that the temperature of the surroundings is 58 o F. If the temperature of the body is now 74 o F, how long ago ( in hours ) was the time of death? Round the answer to the nearest tenth. 69. A kettle full of water is brought to a boil in a room with temperature 22 o C. After min the temperature of the water has decreased from 00 o C to 75 o C. Find the temperature after another min. PAGE 2

13 70. An unknown substance has a hydrogen ion concentration of Find the ph. [ H + ] = M PAGE 3

14 ANSWER KEY. 4 x x (, ) 4. y= 7 5. ( 8, ) x 7. $ $ =3. x=e log 00 0 = log n 5 =z x= 2+log 0.9 e x=3 y=log 2 x 8. y=log ( x) (, 2) ( 2, ) ( 0,) 4,2) log x 3 +log x 9 3 log x 6 log log ( 9 3) log ( 6 x2 +7) 26. 8log x a log y a 6log z a ln ln r 6 9log ( a) 5log ( b) ln ( x)+ ln y ln ( s ) log c 2 ln z 3 2 log ( 0 x5 +9) 2 log 0 x5 + log x 5 + log y 25 + log z 25 5ln ( z)+ ln x 2 4 log log 3 6 ( 7 ) 36. ln 6x 3 x 2 +7 PAGE log x ln y2 +3y+

15 ANSWER KEY 37. log , % , , x ( 6,8) (,3) 6. ( log ( 2 ),log 7 0 ) x ( 5,5) p=000e 0.08t n ( t)=8600e 0.25t m ( t)=85e t PAGE 2