Exponential and Logarithmic Functions -- QUESTIONS -- Logarithms Diploma Practice Exam 2

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1 Eponential and Logarithmic Functions -- QUESTIONS -- Logarithms Diploma Practice Eam

2 Logarithms Diploma Style Practice Eam These are the formulas for logarithms you will be given on your diploma ( 1 ) A= P + i M loga = loga M loga N N log = log M + log N log a b ( MN) log c = log n a a c b a a Use this sheet to record your answers NR NR 3. NR NR 1. NR NR 7. NR Copyright Logarithms Diploma Barry Practice Mabillard, Eam

3 Logarithms Diploma Style Practice Eam 1. The graph of f ( ) = b and the graph of reflections of each other about the line 1 g ( ) =, where b > 0, are b A. y= B. y = b C. = 0 D. y = 0 Use the following information to answer the net question. Equation I log y = log 3 Equation II y = 3 Equation III y = 6 Equation IV ( 6) y = 3. A solution to the equation log3 = 6 could be approimated using technology by graphing equations A. I and III B. I and IV C. II and III D. II and IV 1 3. The epression log A. log5 B. log 1 5 C. log ( 5 ) D. log5 is equivalent to Logarithms Diploma Practice Eam 4

4 Use the following information to answer the net question. The power rating of a particular dynamic electronic circuit is given by the equation P= t w where P is the power rating, t is amount of time since the circuit is switched on, and w is a constant. 4. After the circuit has been operational for 43 seconds, the power rating is The value of w, to the nearest hundredth, is A B. 0.5 C D The epression log ( yz) log ( yz) is equivalent to A. log ( 3 yz) y B. log z C. 3log y + log z log y+ log z D. 1 ( ) 4 6. The value of b in the equation 7= 3+ b is equivalent to A. log B. log4 7 log 3 C. D Numerical Response 1. If y = 8, then to the nearest tenth, the value of 5log + 5log y is. Logarithms Diploma Practice Eam 5

5 The solution to the equation = 5, correct to the nearest hundredth, is A B C D Use the following information to answer the net question. The mass of a radioactive sample is represented in the graph below. The initial mass of 3 mg decays to 8 mg after 1 hours. 8. The half-life of the radioactive sample, in minutes, is A. 60 B. 40 C. 630 D. 160 Numerical Response. 5 a Given ( log ) 3 + a c 8 =, the value of, to the nearest hundredth is. c Logarithms Diploma Practice Eam 6

6 Use the following information to answer the net question. The partial graph of a transformed logarithmic function g() is shown below 9. If the graph of g() is transformed to a new graph h(), and the point (0, a) becomes (a, 0), then a possible transformation is 1 A. h ( ) = g ( ) B. h ( ) = ag ( ) C. h ( ) = g ( ) + a D. 1 h ( ) = g ( ) + a 10. A skilled player at the video game Dot Gobbler has an average high score of points. For every day the player is on vacation, she can epect to lose.7% of her gaming ability. An equation that may be used to predict the average score S of the player after d days is A. S = 50000(.7) 1 d B. S = 50000( 0.07) d C. S = 50000( 0.973) 1 d D. S = 50000( 0.973) d Logarithms Diploma Practice Eam 7

7 Use the following information to answer the net two questions. The decibel level of a sound may be calculated using the formula 1 ( I) L= 10log 10 where L is the loudness of the sound (db) and I is the intensity of the sound. 11. An equation that can be used to solve for the value of I is A. B. C. L I = 10 log10 L I = log I = 10 L I = 10 D. 13 L The loudness of a jet engine is 150 db. The magnitude of the sound intensity is A. 1.5 B C D Numerical Response 3. The epression log b b is equivalent to a numerical value of. Logarithms Diploma Practice Eam 8

8 ( ) The inverse of f = + is A. 1 f ( ) = B. 1 f ( ) = log3 ( 4) C. 1 f ( ) = 4 3 D. 1 f ( ) = 3 log 4 4. The graph of ( ) y 1 =, where b < 0, is the same as the graph of b A. y = log b reflected in the line y = B. y = log b reflected in the line y = C. y = log b reflected in the line y = 0 D. y = b Use the following information to answer the net question. Newton s Law of cooling can be represented by the equation Tt () = Te kt 0 where T(t) is the final temperature in degrees Celsius, T 0 is the initial temperature in degrees Celsius, t is the elapsed time in minutes, and both e & k are constants. e =.718 k = Numerical Response 4. The length of time, in minutes, required for a cup of coffee to cool from 8 C to 65 C is. Logarithms Diploma Practice Eam 9

9 Use the following information to answer the net four questions. The graph of f( ) = 4 is shown below 15. The graph of f( ) = 4 and the graph of g( ) = log4 are symmetrical with respect to the line A. = 0 B. y = 0 C. y = D. y = If the graph of g( ) log4 the new domain of the graph is A. > B. > 3 C. > 4 D. > 1 = undergoes the transformation y g( ) = 3 1 +, 17. A student wis hes to solve the equation 4 = 8. An incorrect procedure to determine the solution is A. Take the logarithm of both sides, use the power rule of logarithms, then isolate the variable by dividing both sides of the equation by log 4. B. Graph y 1 = 4 and y = 8 in a graphing calculator, find the point of intersection, then state the y-value of this point as the solution. C. Draw y 1 = 4 and y = 8 carefully on graph paper, then approimate the coordinates of the point of intersection. D. Find a common base for each side of the equation, then simplify and solve. Logarithms Diploma Practice Eam 10

10 18. The equation 4 y = is represented by graph A. B. C. D. 19. The equation f ( ) 7a + = b, has an intercept equivalent to log b log 7 A. = log a B. y + b C. = 7a D. = 0 = 7a b Logarithms Diploma Practice Eam 11

11 0. A student solves for a in the equation log7 ( 81a) = b. The student determines a is equivalent to the epression A. log81 a = b log B. a = 3 b C. b a = 7 81 D. a = 3 Use the following information to answer the net three questions. The profit of a small business is given below Year Profit $8 000 $3 000 $ $ $ Numerical Response 5. If the owner of the business uses an eponential regression to predict the profits in the year 010, the epected profit (in thousands) is. Numerical Response 6. The business will achieve a profit of $ in the year. 1. A rival business has their profit increase modeled by the function Pt ( ) = 13500(1.4) t. The profits of this business will overtake the profits of the first business, for the first time, in the year A. 0 B. 05 C. 09 D. 030 Logarithms Diploma Practice Eam 1

12 5 4 =. Given the equation a b, an epression for a is A. 4 b 5 4 B. ( b) 5 C. D. 4 b 5 1 ( b) 4 5 Use the following information to answer the net question. A student analyzes the following graph: ( ) = log ( 6 ) f 3. The domain of this graph is A. < 0 B. < 6 C. 0< < 6, 1 D. 1< < 6 4. The -intercept of the graph y = blog c a is A. a 1 B. a C. = 0 D. b Numerical Response 7. Given the equatio n log+ 3log = 8, a student determines the value of to be. Logarithms Diploma Practice Eam 13

13 5. One third of 3 34 is A B C D Given the equ ation log + y= log z, an epression for y is A. y = log a z B. z y = log a C. y = log a z D. y = z ( ) a a 7. The p rice of a vintage video game with the bo and instructions doubles every 0 years. If the video game initially cost $60.00, the value of the game in 33 years will be A B C D The population of a city can be determined using the equation where P is the future population, and t is the time in years. An equation representing t as a function of P is A. P t = B. t = log P 5 log1.03 C. log P t = 5log1.03 D. log P 5 t = log1.03 P = (1.03) t Logarithms Diploma Practice Eam 14

14 ( ) ( ) 9. The value of in the equation log + log + = log 3 is A. = -1 B. = 1 C. =±1 D. No Solution 30. If log6 = 10, then log6 A. 0.5 B C. 84 D is ( )( ) logbc logbc 31. An epre ssion equivalent to a a is A. a + a B. C. D. logb c ( a ) log b c logb a c b logc a 3. The graph of y = b, where b < 1, undergoes the transformation y+ 3 = f( ). The range of the transformed graph is A. y < 3 B. y > 3 C. y 3 D. y 3 ( ) ( ) 33. The solutio n of log + + log 1 = 1 is A. -4 B. 3 C. -4, 3 D. No Solution Logarithms Diploma Practice Eam 15

15 Written Response 10% ( ) = log( + ) 1. Draw the graph of f in the space provided below Complete the following chart to describe the graph above Domain Range Equation of Asymptote -intercept y-intercept y-value when = Describe the transformations app lied to the graph of y = log in order to obtain the graph of f ( ) = log( + ) The domain of the general epression f ( ) = alog( b+ c) + d is Logarithms Diploma Practice Eam 16

16 Use the following information to answer the net question. A student is asked to solve the equation techniques learned in Pure Math = 43 using different Written Response 10%. Eplain how to solve the equation graphically. Indicate appropriate window settings and state the solution. Algebraically show how to solve this equation using a common base. Algebraically show how to solve this equation by taking the logarithm of both sides and solving for. The student is now asked to solve the equation log3 = 4. Eplain how this can be done with a graphing calculator. Logarithms Diploma Practice Eam 17

17 Use the following information to answer the net question. A useful equation for solving application questions is 0 ( ) A= A b where A is the future amount, A 0 is the initial amount, b is the rate of growth or decay, P is the period, and t is the elapsed time. t P Written Response 10% 3. Algebraically solve for P A particular bacteria doubles every P hours. If a bacterial culture starts with bacteria and has bacteria after 3 hours, determine the doubling period. The population of a town triples every 8 years. Determine the number of years it will take for the population to double. Logarithms Diploma Practice Eam 18

18 Light passing through dirty water retains only ¾ of its intensity for every metre of water. Determine the depth at which the light will have 64% of its surface intensity. A particular painting goes down in value by 4.3% each year due to improper storage. Determine the number of years it will take for the value of the painting be half the initial worth. You have now completed the eamination. Please check over your answers carefully before self-marking. Good luck on your real eam! Logarithms Diploma Practice Eam 19