Welcome to the Webinar “Live” Review for Final Exam (MAC 1105) We will start promptly at 8:00 pm Everyone will be placed on mute I am happy that you are able to join us 

Important Note The following slides present sample problems for Chapter 6 only. Use the slides from Tests1-3 to review the previous material.   To be prepared for the test, you must review the material from all the assigned sections throughout the term.

Which of the following is the graph of f(x) = (1/3)x ? (Each tic-mark is one unit.)    a. b. For f(x) = bx b > 1, function is increasing; 0 < b < 1, function is decreasing (Review definition of exponential function in section 6.1) Answer: b

The range of the function f(x) = 3x is:   a. x can be any real number b. y can be any real number c. x > 0 d. y > 0 Domain: Possible values of the input. Range: Possible values of the output. Base b for the exponential function: b > 0 and b  1 So, we know that bx > 0. Answer is d. Confirm with graph:

State any transformations for f(x) = 2x+2 – 1, the horizontal asymptote, and the y-intercept. Horizontal Translation: left 2 units Vertical Translation: down 1 unit Horizontal Asymptote: Since graph was shifted 1 unit down, the horizontal asymptote is y = -1. y-intercept  let x = 0 and solve for y f(0) = 2(0+2) – 1 = 22 – 1 = 3

Without graphing nor performing calculations, determine if the function p(t) = 8(1.7)t is increasing or decreasing, and find its vertical intercept. f(x) = cbx b = 1.7 Since b > 1, this function is increasing The y-intercept of f(x) = cbx is (0, c). Since c = 8, the vertical intercept is (0, 8)

[Make sure to distribute correctly: 2(2x + 3) = 4x + 6] Solve 42x+3 = 8, by equating the bases. 42x+3 = 8 (22)2x+3 = 23 24x+6 = 23 [Make sure to distribute correctly: 2(2x + 3) = 4x + 6] Solving the equation: 4x + 6 = 3 x = -3/4

logk r = t is equivalent to kr = t   True b. False Hint: Start at the base and “move counterclockwise” logk r = t logk r = t is equivalent to kt = r Answer: False Numeric verification log5 25 = 2 is equivalent to 52 = 25

Solve algebraically for x : logx 4 = 1/2 In exponential form: x1/2 = 4 We know x1/2 is equivalent to So, x = 16 because (16)1/2 = 4     

The x-intercept implies that y = 0. So, we have logb x = 0 If y = logb x, for any base b (b > 0, b  1) the x-intercept is (1, 0).   a. True b. False The x-intercept implies that y = 0. So, we have logb x = 0 In exponential form: b0 = x By definition, for any number a not equal to 0, a0 = 1. Therefore, if b0 = x, then x = 1 So, we have (1, 0) Answer: True (Review definition of Logarithmic Function in Section 6.2.)

Simplify without a calculator: log3 3 a. 0 b. 1 c. 3 d. Undefined Applying the basic property logb b = 1, we know the answer is b. Verifying: log3 3 = y means that 3y = 3, thus y must equal 1.

Your time to work! Poll Question log mn is equivalent to m log n (log m)(log n) (log m)/(log n) Your time to work! Poll Question

log mn is equivalent to m log n n log m (log m)(log n) (log m)/(log n) Answer: b (power property)

True or False: The following is the graph of y = log (x – 4) (Each tic-mark is one unit.)  y = log (x – 4) implies a horizontal translation 4 units right; not a vertical translation. Answer: False Graph of y = log(x) is Answer should have been

True or False: The function y = log8x is the inverse function of y = x8 . Answer: False The inverse function of y = log8x is y = 8x . y = log8x x = log8y log8y = x 8x = y Observe that in y = x8 the exponent is not a variable!

Hint: What formula is helpful here? Use your graphing calculator to evaluate log864. Round to 3 decimal places where needed. 0.5 0.903 1.631 2 Your time to work! Poll Question Hint: What formula is helpful here?

Use your graphing calculator to calculate log864 Use your graphing calculator to calculate log864. Round to 3 decimal places where needed. 0.5 0.903 1.631 2 Answer: 2 log864 = log(64) log (8)

ln e is equivalent to logee = 1 Solve algebraically for x: 3e2x + 3 = 12 3e2x + 3 = 12 e2x + 3 = 4 ln e2x + 3 = ln 4 (2x + 3) ln e = ln 4 ln e is equivalent to logee = 1 2x + 3 = ln 4 x = ln 4 – 3 2  

True or False: ln (1/e-q) = q 1/e-q is equivalent to eq Therefore, = ln (eq) = q ln (e) ln (e) is equivalent to logee = 1 = q(1) = q True

Solve the equation ln (2x – 1) = 3 and approximate your answer to 4 decimal places.   50.5000 b. 27.2991 c. 10.5428 d. 5.7708 ln (2x – 1) = 3 This is equivalent to loge(2x – 1) = 3 In exponential form, e3 = 2x – 1 2x = e3 + 1 2x = 21.085537 x = 10.542768  10.5428 Answer is c.

Rewrite as a single logarithm: logb (m) + 5 logb (n) - logb (r) = logb (mn5) - logb (r) = logb

Solve the equation log (x) + log (x + 3) = log(5x + 3).   log [(x)(x + 3)] = log(5x + 3) log (x2 + 3x) = log(5x + 3) By Logarithmic Equality, x2 + 3x = 5x + 3 Solving for x: x2 + 3x – 5x – 3 = 0 x2 – 2x – 3 = 0 (x – 3)(x + 1) = 0 x = 3; –1 Checking for extraneous solutions, discard x = –1 (Remember domain of log (x) > 0) Answer: x = 3

Suki bought a computer for $500 Suki bought a computer for $500. She estimates that the computer will depreciate so that each year its value will be ¾ of its worth for the previous year. Write a function V(t) that models this situation, where t represents number of years. Find Suki’s computer value after 2 years. If the current value of this computer is $160, approximate its age in years. Exponential growth and decay: P(t) = P0bt The initial value, V0 = 500, decay factor = ¾, so V(t) = 500(3/4)t b. V(2) = 500(3/4)2 = $281.25

c. If the current value of this computer is $160, approximate its age in years. We know V(t) = 500(3/4)t Solving Algebraically: 500(3/4)t = 160 (3/4)t = 160/500 (3/4)t = 0.32 ln(3/4)t = ln 0.32 t ln(3/4) = ln 0.32 t = ln 0.32 ln(3/4) t = 3.96  4 years

c. If the current value of this computer is $160, approximate its age in years. Solving by Graphing: 500(3/4)t = 160 Y1 = 500(3/4)x Y2 = 160 t = 3.96  4 years Caution: You need to find an appropriate window!

Jeremy invested $1,000 in an account paying 1. 5% compounded monthly Jeremy invested $1,000 in an account paying 1.5% compounded monthly. Find the accumulated amount after 5 years. A = where P = 1000, r = 0.015, n = 12, and t = 5. A  $1077.83 (This formula will be provided on the test ) Careful with parentheses!

NOTE: It is important that you follow the instructions on the test. For example, if asked for interval notation, write your answer using interval notation. If you are not asked for interval notation, then just write the values separated by commas (,) or semicolons (;) as specified on the question. If asked to write log(x) with parentheses, do not just write log x. So, read and follow instructions!

Good luck!  Do well! gacosta@valenciacollege.edu