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

2. 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

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

4. 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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. ConcepTest • Section 1.4 • Question 6

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

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

25. 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

26. 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

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

28. 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

29. 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

30. 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