HW#1. Unit 4B Logarithmic Functions HW #1. 1) Which of the following is equivalent to y=log7 x? (1) y =x 7 (3) x = 7 y (2) x =y 7 (4) y =x 1/7

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1 HW#1 Name Unit 4B Logarithmic Functions HW #1 Algebra II Mrs. Dailey 1) Which of the following is equivalent to y=log7 x? (1) y =x 7 (3) x = 7 y (2) x =y 7 (4) y =x 1/7 2) If the graph of y =6 x is reflected across the line y = x then the resulting curve has an equation of (1) y = -6 x (3) x = log6 y (2) y =log6 x (4) x = y 6 3) The value of log5 167 is closest to which of the following? Hint guess and check the answers. (1) 2.67 (3) 4.58 (2) 1.98 (4) ) Which of the following represents the y-intercept of the function y = log(x +1000)- 8? (1) -8 (3) 3 (2) -5 (4) 5 y-intercept: x = 0 5) Write as an exponential equation, then evaluate each. (Easy) (a) log2 32 (b) log7 49 (c) log (d) log4 1024

2 6) Write as an exponential equation, then evaluate, (Medium) (a) log2 (b) log5 (c) log ) Write as an exponential equation, get like bases, and then evaluate. Each of these will have non-integer, fractional answers. (Difficult) 3 (a) log4 2 (b) log 5 5

3 HW#2 Name Unit 4B Logarithmic Functions HW #2 Algebra II Mrs. Dailey 1) The domain of y= log3(x+5) in the real numbers is (1) {x x > 0} (3) {x x > 5} (2) {x x >-5} (4) {x x -4} 2) Which of the following equations describes the graph shown below? (1) y = log5 x (3) y = log3 x (2) y =log2 x (4) y = log4 x 3) Which of the following represents the y-intercept of the function y=log2(32-x)-1? (1) 8 (3) -1 (2) -4 (4) 4 4) Which of the following values of x is not in the domain of f(x) = log5(10-2x)? (1) -3 (3) 5 (2) 0 (4) 4

4 5) Which of the following is true about the function y=log4(x+16) -1? (1) It has an x-intercept of 4 and a y-intercept of -1. (2) It has x-intercept of -12 and a y-intercept of 1. (3) It has an x-intercept of -16 and a y-intercept of 1. (4) It has an x-intercept of -16 and a y-intercept of -1. 6) Determine the domains of each of the following logarithmic functions. State your answers using any accepted notation. Be sure to show the inequality that you are solving to find the domain and the work you use to solve the inequality. (a) y =log5(2x-1) (b) y =log(6-x) 7) For what value of x will log2 x =10? For what value of x will log2 x = 20?

5 HW#3 Name Unit 4B Logarithmic Functions HW #3 Algebra II Mrs. Dailey 1) Solve for x, to the nearest hundredth: 7 x =500 x 3 2) Solve for x, to the nearest hundredth: ) Solve for x to the nearest hundredth: 5 x-4 = 275 is 4) Solve for x, round to the nearest hundredth. (a) 50(2) x = ) Isolate exponential 2) Take log of both sides 3) Solve for x.

6 5) Solve each of the following exponential equations. Be careful with your use of parentheses. Express each answer to the nearest hundredth. (a) 6 2x-5 = 300 6) The population of Red Hook is growing at a rate of 3.5% per year. If its current population is 12,500, in how many years will the population exceed 20,000? Round your answer to the nearest year. Only an algebraic solution is acceptable. A= P(1±r) t 7) Write as an exponential equation, get like bases, and then evaluate. Each of these will have non-integer, fractional answers. (Difficult) 5 (a) log4 8 (b) log 4 2

7 8) A radioactive substance is decaying such that 2% of its mass is lost every year. Originally there were 50 kilograms of the substance present. (a) Write an equation for the amount, A, of the substance left after t-years. A= P(1±r) t (b) Find the amount of time that it takes for only half of the initial amount to remain. Round your answer to the nearest tenth of a year. 9) Find the solution to the general exponential equation a(b) cx =d, in terms of the constants a, c, d and the logarithm of base b. Think about reversing the order of operations in order to solve for x.

8 HW#4 Name Unit 4B Logarithmic Functions HW #4 Algebra II Mrs. Dailey 1) Which of the following is closest to the y-intercept of the function whose equation is y =10e x+1? (1) 10 (3) 27 y-intercept: x= 0 (2) 18 (4) 52 2) On the grid below, the solid curve represents y =e x. Which of the following exponential functions could describe the dashed curve? Explain your choice. x (1) y 1 2 (3) y = 2 x (2) y = e -x (4) y = 4 x 3) Which of the following values of t solves the equation 5e 2t = 15? (1) ln15 (3) 2ln (2) (4) ln 3 2ln 5 2 4) At which of the following values of x does f(x) =2e 2x -32 have a zero? 5 (1) ln (3) ln 8 2 (2) ln 4 (4) 2 y ln 5 zeros: y = 0

9 5) For the equation ae ct = d, solve for the variable t in terms of a, c, and d. Express your answer in terms of the natural logarithm. 6) Flu is spreading exponentially at a school. The number of new flu patients can be modeled using the equation F =10e 0.12d, where d represents the number of days since 10 students had the flu. How many days will it take for the number of new flu patients to equal 50? Determine your answer algebraically using the natural logarithm. Round your answer to the nearest day. 7) Solve for x, round to the nearest hundredth: x ) Isolate exponential 2) Take log of both sides 3) Solve for x.

10 HW#5 Name Unit 4B Logarithmic Functions HW #5 Algebra II Mrs. Dailey 1) Which of the following is not equivalent to log 36? (1) log 2 + log 18 (3) log 30 + log 6 (2) 2log 6 (4) log 4 + log 9 2) The log3 20 can be written as (1) 2log3 2 + log3 5 (3) log3 15+ log3 5 (2) 2log3 10 (4) 2log log ) Which of the following is equivalent to log x? 3 y (1) log x - log y 1 (3) 3log x log y 3 (2) 9log(x-y) y (4) log(3 x) log 3 4) The difference log2 3 - log2 12 is equal to (1) -2 (3) 1/4 (2) -1/2 (4) 4

11 5) If log5 = p and log 2 = q then log 200 can be written in terms of p and q as (1) 4p + q (3) 2(p + q) (2) 2p + 3q (4) 3p + 2q 6) When rounded to the nearest hundredth, log3 7 =1.77. Which of the following represents the value of log3 63 to the nearest hundredth? Hint: Rewrite 63 as a product involving 7. (1) 3.54 (3) 3.77 (2) 8.77 (4) ) The expression 4 log x log y 3log z can be rewritten equivalently as (1) log xz 4 3 xz (3) log y 2y 4 3 6xz 6xz (2) log (4) log y y 8) If k =log2 3, then log2 48 = (1) 2k + 3 (3) k + 8 (2) 3k +1 (4) k + 4

12 9) If g(x) = 8x 6 and f(x) = log4 (2x) then f(g(x)) =? (1) 4log4x + 1 (3) 2(3log4x+1) (2) 3(log4x + 2) (4) 6 log4 x + 4 e 10) The logarithmic expression ln 3 can be rewritten as y (1) 3ln y - 2 (3) ln y 6 2 (2) 1 6ln y 2 (4) ln y 3 11) The savings in a bank account can be modeled using S=1250e.045t, where t is the number of years the money has been in the account. Determine, to the nearest tenth of a year, how long it will take for the amount of savings to double from the initial amount deposited of $1250.

13 HW#7 Name Unit 4B Logarithmic Functions HW #7 Algebra II Mrs. Dailey 1) Solve for x: log 2(2x 3) 3 2) For the function f(x) = a(b) x, where 0 < b <1, describe the behavior of the given function as x approaches infinity. 3) For the function g(x) = a(b) x + k, where 0 < b <1,, describe the behavior of the given function as x approaches infinity. 4) Given f(t) = a(b) t + c, which of the following represents the y-intercept of this function? Show how you arrived at your choice. (1) a (3) c (2) a + c (4) b + c

14 5) Graph y = log2 (x+3) 5 on the set of axes below. Use an appropriate scale to include both intercepts. (June 17) Describe the behavior of the given function as x approaches -3 and as x approaches positive infinity.

15 HW#8 Name Unit 4B Logarithmic Functions HW #8 Algebra II Mrs. Dailey 1) A liquid starts at a temperature of 175 ºC and cools down in a room held at a constant temperature of 16 ºC. It's temperature can be modeled, as a function of time cooling, by the equation y = a(b) x +c. Which of the following statements is true? (1) a =159 and c =16 (3) a =175 and c =16 (2) a =16 and c =159 (4) a =16 and c =175 2) A cooling liquid has a temperature given by the function T(m) = 132(.83) m + 40, where m is the number of minutes it has been cooling. Which of the following temperatures did the liquid start at? (1) 40 (3) 172 (2) 92 (4) 132 3) A liquid starts at a temperature of 190 ºF and cools down in a room held at a constant 65 ºF. After 10 minutes of cooling, it is at a temperature of 92 ºF. The Fahrenheit temperature, F, can be modeled as a function of time in minutes, t, by the equation: F(t) = 125(0.86) t + 65 Algebraically, determine the number of minutes it will take for the temperature to reach 70 ºF. Round to the nearest tenth of a minute.

16 4) A Warming Liquid - A liquid is taken out of a refrigerator and placed in a warmer room, where its temperature, in F, increases over time. It can be modeled using the equation T(m) = 74-39(0.87) m. (a) What temperature did the liquid start at? Show the work that leads to your answer.

17 Unit 4B Review Name CC Algebra II Unit 4B Logarithms Date Mrs. Dailey Notes 1) Argument of the logarithm must be greater than 0 (Question #: 2,11,12,17) Example: log3 (3x-6) Argument is 3x -6 Domain of function is when 3x-6 > 0 3x >6 Domain: x > 2 Cannot take a logarithm of a negative number or zero. log3 (-3) There is no number that could satisfy 3 x = -3 2) Working with negative exponents: 1 x 3 x ) Change of Base Formula: (Question #: 4,5) Example: log2 32 log(32) 5 log(2) 4) Find x-intercept and y-intercept. (Question #: 26) x-intercept set y= 0 y-intercept set x=0 5) Properties of Logarithms: (Question #: 6,14,15,21,24,25) (1) Log (2) Logs Product Property logb MN logb M logb N M Quotient Property: logb logb M logb N N x Power Property: log M x log M b b 6) Rewrite and solve exponential equations using logarithms. (Question #: 1,7,8,10,13,16,22,23) Exponential to log log to exponential

18 7) Continuous Compound A = Pe rt (Question #: 9,18,21) P: Initial Amt r: Percent as decimal (negative if decreasing) t: Time (based on the percent: years if annual %, months if monthly percent)

19 Unit 4B Review Name CC Algebra II Unit 4B Logarithms Date Mrs. Dailey 1 1) If a 0 then log a 5 a is equal to (1) (2) (3) -5 5 (4) 5 2) Given the function f x log 2x 8 domain of the function? (1) x 5 (3) x 8 (2) x 2 (4) x 20, which of the following values of x is not in the 2 3) The temperature of a cooling liquid in a room held at a constant 75 degrees Fahrenheit can be described by the equation t F t , where F is the Fahrenheit temperature and t is the amount of time it has been cooling, in minutes. Which of the following was the original temperature of the liquid when it began cooling? (1) 75 (3) 203 (2) 132 (4) 207 4) Which of the following is closest to the value of log4 40? (1) 1.8 (3) 2.7 (2) 2.3 (4) 3.5 5) Which of the following equations is shown graphed on the grid below? x (1) y 2 2 x (2) y 4 2 (3) y log x 2 (4) y log x Asymptote x=2

20 6) Which of the following is equivalent to log x 2 y? (1) log x 2 log y (3) 2log x log y 2 (2) 2log x log y (4) 4log x log y 2 7) If x b then x = (1) log 7 b 5 3 (2) 5 b log 1.4 (3) (4) 7 5 3b b 8) If f x 90? x f x then which of the following values of x solves the equation (1) 12.1 (2) 14.4 (3) 15.8 (4) 18.3

21 9) How many years, to the nearest tenth, would it take for a $500 investment to double if it is earning a continuous compound interest of 3.5% per year? (1) 17.4 years (3) 22.5 years (2) 19.8 years (4) 25.1 years kt 10) If a liquid is cooling down according to the formula y 84e 55 and at t = 22 the temperature is y = 71 then which of the following is the value of k to the nearest hundredth? (1) 0.08 (3) 0.29 (2) 0.27 (4) ) On the grid shown below, the graph of f x 2 x is shown. (a) On the same graph grid, create an accurate sketch of this 1 function s inverse, f x. (b) State the equation of f 1 x. 1 f x and f x 1 f x f x (c) State the domain and range of both. Domain: Range: Domain: Range:

22 12) For the logarithmic function f x log x 4 domain., explain why x = 2 is not in its , where w represents the worth in dollars and t represents the number of years since the principal was deposited into the account. Algebraically determine the number of years, to the nearest tenth, it takes for the account to be worth $ ) A bank account s worth can be modeled using the formula wt 4t 14) The expression 2logx (3logy + log z) is equivalent to (June 10) 1) 2) 3) 4)

23 15) If logx 2 log2a = log3a, then log x expressed in terms of log a is equivalent to (Jan 12) 1) 2) 3) 4) log 4 16) Solve algebraically for x: 5x1 (June 14) 1 3

24 17) If a function is defined by the equation f(x) = 4 x, which graph represents the inverse of this function? (Fall 09) (1) (3) (2) (4) 18) The number of bacteria present in a Petri dish can be modeled by the function N=50e 3t, where N is the number of bacteria present in the Petri dish after t hours. Using this model, determine, to the nearest hundredth, the number of hours it will take for N to reach 30,700. (Jan 13)

25 19) Solve for x: log 2(2x 3) 3 20) Evaluate e xlny when x=3 and y=2. (June 11) ln e 1 21) Solve for x to the nearest thousandth. 0.05x 150e 340

26 22) Solve for x, round to the nearest hundredth. 6(2) x+3 = ) Solve algebraically for x: log 27(2x 1) (June 13) 3

27 9 24) If a= log 2 and b= log 3, the expression log is equivalent to (Jan 13) 20 1) 2b a + 1 2) 2b - a -1 3) b 2 a ) 2b a 1 25) If a= log 3 and b= log 2, rewrite log 12 using a and b. 26) The x-value of which function s x-intercept is greater. Justify your answer. f(x) = log (3x-4) g(x) = 2x - 3