ConcepTest • Section 1.4 • Question 1 Which is a graph of y = ln x? ConcepTest • Section 1.4 • Question 1

ConcepTest • Section 1.4 • Answer 1 (c). y = ln x is an increasing function passing through the point (1, 0). COMMENT: You could discuss which properties each of the remaining graphs possess that made the student conclude that it was not an appropriate choice. Follow-up Question. Find possible formulas for the remaining graphs. Answer. (a) y = x2, (b) y = ex, and (d) y = – ln x. ConcepTest • Section 1.4 • Answer 1

ConcepTest • Section 1.4 • Question 2 The graph in Figure 1.18 could be that of ConcepTest • Section 1.4 • Question 2

ConcepTest • Section 1.4 • Answer 2 (c). Note that (a) is the graph of ln x shifted up ½, (b) is shifted down ½, (d) is shifted to the right ½. COMMENT: You could also distinguish the four graphs by their horizontal intercepts. ConcepTest • Section 1.4 • Answer 2

ConcepTest • Section 1.4 • Question 3 Which of the following functions have vertical asymptotes of x = 3? ConcepTest • Section 1.4 • Question 3

ConcepTest • Section 1.4 • Answer 3 (b). Note that (a) and (d) have vertical asymptotes at x = 0, while (c) has one at x = –3, and (b) has one at x = 3, as desired. COMMENT: Follow-up Question. Do any of these functions have horizontal asymptotes? If so, what are they? Answer. No, the range of these functions is all real numbers. ConcepTest • Section 1.4 • Answer 3

ConcepTest • Section 1.4 • Question 4 Without calculating the following quantities, Use properties of logarithms to decide which of the following is largest. ConcepTest • Section 1.4 • Question 4

ConcepTest • Section 1.4 • Answer 4 (b). ln(30) – ln(2) = ln(15), 2ln(4) =ln(16), ln(3) + ln(4) = ln(12), and Since e2 < 9 and ln x is an increasing function, ln(16) is the largest number. COMMENT: Point out that using the rules of logarithms enables us to compare exact values. If the comparison were made using a computer or calculator, we would likely be comparing approximate values. ConcepTest • Section 1.4 • Answer 4

ConcepTest • Section 1.4 • Question 5 The graph of a logarithmic function has a horizontal asymptote. (a) True (b) False ConcepTest • Section 1.4 • Question 5

ConcepTest • Section 1.4 • Answer 5 (b). The range of logarithmic functions consists of all real numbers. COMMENT: You could also ask about vertical asymptotes. ConcepTest • Section 1.4 • Answer 5

ConcepTest • Section 1.4 • Question 6

ConcepTest • Section 1.4 • Answer 6 (d) COMMENT: Follow-up Question. What is the value of Answer. ln (M + N). ConcepTest • Section 1.4 • Answer 6

ConcepTest • Section 1.4 • Question 7 If log10(x – a) = n, then x = (a) 10a+n (b) a + 10n (c) n + 10a (d) n + a10 ConcepTest • Section 1.4 • Question 7

ConcepTest • Section 1.4 • Answer 7 (b). Compose each side with the exponential function 10x since it is the inverse function of log10 x. COMMENT: You could ask the same question with the natural logarithm rather than the logarithm base 10. ConcepTest • Section 1.4 • Answer 7

ConcepTest • Section 1.4 • Question 8 Which of the following functions are increasing and concave up? (a) 3–x (b) 3x (c) ln x (d) – ln x ConcepTest • Section 1.4 • Question 8

ConcepTest • Section 1.4 • Answer 8 (b). Note that (a) and (d) are decreasing, and (c) is concave down. COMMENT: You could also ask about asymptotes (horizontal and vertical) and intercepts for all four functions. ConcepTest • Section 1.4 • Answer 8

ConcepTest • Section 1.4 • Question 9 Which of the following functions are decreasing and concave up? (a) – ln(4 + x) (b) 3x – 4 (c) 34 – x (d) ln(4 – x) ConcepTest • Section 1.4 • Question 9

ConcepTest • Section 1.4 • Answer 9 (a) and (c). Note that (b) is increasing and (d) is concave down. COMMENT: You could also ask about asymptotes (horizontal and vertical) and intercepts for all four functions. ConcepTest • Section 1.4 • Answer 9

ConcepTest • Section 1.4 • Question 10 Which of the following does not have a horizontal asymptote? (a) y = ln x (b) y = 1/x (c) y = 5x (d) y = x1/3 ConcepTest • Section 1.4 • Question 10

ConcepTest • Section 1.4 • Answer 10 (a) and (d). The range of log x and x1/3 is all real numbers. COMMENT: Follow-up Question. Which of the above functions does not have any asymptotes? Answer. (d). The domain and range of y = x1/3 is all real numbers. ConcepTest • Section 1.4 • Answer 10

ConcepTest • Section 1.4 • Question 11 Give a formula for the inverse of the following function: P = f (t) = 16e14t. ConcepTest • Section 1.4 • Question 11

ConcepTest • Section 1.4 • Answer 11 (d). If P = 16e14t, then P/16 = e14t and ln (P/16) = 14t. This gives t = 1/14 ln (P/16). COMMENT: Students may find that f -1 (P) = 1/14 (ln P – ln 16). This would be an excellent time to review the properties of logarithms. For an alternate question, you could also use f (t) = 2 – e -3t. ConcepTest • Section 1.4 • Answer 11

ConcepTest • Section 1.4 • Question 12 Give a formula for the inverse of the following function: P = f (t) = 16 ln(14t). ConcepTest • Section 1.4 • Question 12

ConcepTest • Section 1.4 • Answer 12 COMMENT: You could also use f (t) = 6 + 2 ln(3t – 1). ConcepTest • Section 1.4 • Answer 12

ConcepTest • Section 1.4 • Question 13 Solve for x if 8y = 3ex. (a) x = ln 8 + ln 3 + ln y (b) x = ln 3 – ln 8 + ln y (c) x = ln 8 + ln y – ln 3 (d) x = ln 3 – ln 8 – ln y ConcepTest • Section 1.4 • Question 13

ConcepTest • Section 1.4 • Answer 13 COMMENT: This is a good place to point out the many ways answers can be expressed using logarithms. ConcepTest • Section 1.4 • Answer 13

ConcepTest • Section 1.4 • Question 14 Solve for x if y = e + 2x. ConcepTest • Section 1.4 • Question 14

ConcepTest • Section 1.4 • Answer 14 COMMENT: You could ask what errors could have been made in obtaining the other choices. ConcepTest • Section 1.4 • Answer 14

ConcepTest • Section 1.4 • Question 15 For what value of x is 3 · 3–x + 4 = 16 – 3–x? ConcepTest • Section 1.4 • Question 15

ConcepTest • Section 1.4 • Answer 15 If 3 · 3–x + 4 = 16 – 3–x, then (3 + 1)3–x = 16 – 4 = 12. Division gives 3–x = 3, so –x = 1 and x = –1. COMMENT: Students may try to take logarithms of both sides of the original equation. ConcepTest • Section 1.4 • Answer 15