6.3E: Exercises

Exercises

In Exercises 1 - 33, solve the equation analytically.

1. \(2^{4x} = 8\)
2. \(3^{(x - 1)} = 27\)
3. \(5^{2x-1} = 125\)
4. \(4^{2x} = \frac{1}{2}\)
5. \(8^{x} = \frac{1}{128}\)
6. \(2^{(x^{3} - x)} = 1\)
7. \(3^{7x} = 81^{4-2x}\)
8. \(9 \cdot 3^{7x} = \left(\frac{1}{9}\right)^{2x}\)
9. \(3^{2x} = 5\)
10. \(5^{-x} = 2\)
11. \(5^{x} = -2\)
12. \(3^{(x - 1)} = 29\)
13. \((1.005)^{12x} = 3\)
14. \(e^{-5730k} = \frac{1}{2}\)
15. \(2000e^{0.1t} = 4000\)
16. \(500\left(1-e^{2x}\right) = 250\)
17. \(70 + 90e^{-0.1t} = 75\)
18. \(30-6e^{-0.1x}=20\)
19. \(\dfrac{100e^{x}}{e^{x}+2}=50\)
20. \(\dfrac{5000}{1+2e^{-3t}}=2500\)
21. \(\dfrac{150}{1 + 29e^{-0.8t}} = 75\)
22. \(25\left(\frac{4}{5}\right)^{x} = 10\)
23. \(e^{2x} = 2e^{x}\)
24. \(7e^{2x} = 28e^{-6x}\)
25. \(3^{(x - 1)} = 2^{x}\)
26. \(3^{(x - 1)} = \left(\frac{1}{2}\right)^{(x + 5)}\)
27. \(7^{3+7x} = 3^{4-2x}\)
28. \(e^{2x} - 3e^{x}-10=0\)
29. \(e^{2x} = e^{x}+6\)
30. \(4^{x} + 2^{x} = 12\)
31. \(e^{x}-3e^{-x}=2\)
32. \(e^{x}+15e^{-x}=8\)
33. \(3^{x}+25\cdot3^{-x}=10\)

In Exercises 34 - 39, solve the inequality analytically.

34. \(e^{x} > 53\)
35. \(1000\left(1.005\right)^{12t} \geq 3000\)
36. \(2^{(x^{3} - x)} < 1\)
37. \(25\left(\frac{4}{5}\right)^{x} \geq 10\)
38. \(\dfrac{150}{1 + 29e^{-0.8t}} \leq 130\)
39. \(\vphantom{\dfrac{150}{1 + 29e^{-0.8t}}} 70 + 90e^{-0.1t} \leq 75\)

In Exercises 40 - 45, use your calculator to help you solve the equation or inequality.

40. \(2^{x} = x^2\)
41. \(e^{x} = \ln(x) + 5\)
42. \(e^{\sqrt{x}} = x + 1\)
43. \(e^{-x} - xe^{-x} \geq 0\)
44. \(3^{(x - 1)} < 2^{x}\)
45. \(e^{x} < x^{3} - x\)
46. Since \(f(x) = \ln(x)\) is a strictly increasing function, if \(0 < a < b\) then \(\ln(a) < \ln(b)\). Use this fact to solve the inequality \(e^{(3x - 1)} > 6\) without a sign diagram. Use this technique to solve the inequalities in Exercises 34 - 39. (NOTE: Isolate the exponential function first!)
47. Compute the inverse of \(f(x) = \dfrac{e^{x} - e^{-x}}{2}\). State the domain and range of both \(f\) and \(f^{-1}\).
48. In Example 6.3.4, we found that the inverse of \(f(x) = \dfrac{5e^{x}}{e^{x}+1}\) was \(f^{-1}(x) = \ln\left(\dfrac{x}{5-x}\right)\) but we left a few loose ends for you to tie up.
    1. Show that \(\left(f^{-1} \circ f\right)(x) = x\) for all \(x\) in the domain of \(f\) and that \(\left(f \circ f^{-1}\right)(x) = x\) for all \(x\) in the domain of \(f^{-1}\).
    2. Find the range of \(f\) by finding the domain of \(f^{-1}\).
    3. Let \(g(x) = \dfrac{5x}{x+1}\) and \(h(x) = e^{x}\). Show that \(f = g \circ h\) and that \((g \circ h)^{-1} = h^{-1} \circ g^{-1}\). (We know this is true in general by Exercise 31 in Section 5.2, but it’s nice to see a specific example of the property.)
49. With the help of your classmates, solve the inequality \(e^{x} > x^{n}\) for a variety of natural numbers \(n\). What might you conjecture about the “speed” at which \(f(x) = e^{x}\) grows versus any polynomial?

Answers

1. \(x = \frac{3}{4}\)
2. \(x = 4\)
3. \(x=2\)
4. \(x = -\frac{1}{4}\)
5. \(x = -\frac{7}{3}\)
6. \(x = -1, \, 0, \, 1\)
7. \(x = \frac{16}{15}\)
8. \(x=-\frac{2}{11}\)
9. \(x = \frac{\ln(5)}{2\ln(3)}\)
10. \(x = -\frac{\ln(2)}{\ln(5)}\)
11. No solution.
12. \(x = \frac{\ln(29) + \ln(3)}{\ln(3)}\)
13. \(x = \frac{\ln(3)}{12\ln(1.005)}\)
14. \(k = \frac{\ln\left(\frac{1}{2}\right)}{-5730} = \frac{\ln(2)}{5730}\)
15. \(t=\frac{\ln(2)}{0.1} = 10\ln(2)\)
16. \(x=\frac{1}{2}\ln\left(\frac{1}{2}\right) = -\frac{1}{2}\ln(2)\)
17. \(t = \frac{\ln\left(\frac{1}{18}\right)}{-0.1} =10 \ln(18)\)
18. \(x=-10\ln\left(\frac{5}{3}\right) = 10\ln\left(\frac{3}{5}\right)\)
19. \(x=\ln(2)\)
20. \(t=\frac{1}{3}\ln(2)\)
21. \(t = \frac{\ln\left(\frac{1}{29}\right)}{-0.8} = \frac{5}{4}\ln(29)\)
22. \(x = \frac{\ln\left(\frac{2}{5}\right)}{\ln\left(\frac{4}{5}\right)} = \frac{\ln(2)-\ln(5)}{\ln(4) - \ln(5)}\)
23. \(x = \ln(2)\)
24. \(x = -\frac{1}{8} \ln\left(\frac{1}{4} \right) = \frac{1}{4}\ln(2)\)
25. \(x = \frac{\ln(3)}{\ln(3) - \ln(2)}\)
26. \(x = \frac{\ln(3) + 5\ln\left(\frac{1}{2}\right)}{\ln(3) - \ln\left(\frac{1}{2}\right)} = \frac{\ln(3)-5\ln(2)}{\ln(3)+\ln(2)}\)
27. \(x = \frac{4 \ln(3) - 3 \ln(7)}{7 \ln(7) + 2 \ln(3)}\)
28. \(x=\ln(5)\)
29. \(x=\ln(3)\)
30. \(x=\frac{\ln(3)}{\ln(2)}\)
31. \(x=\ln(3)\)
32. \(x=\ln(3)\), \(\ln(5)\)
33. \(x=\frac{\ln(5)}{\ln(3)}\)
34. \((\ln(53), \infty)\)
35. \(\left[\frac{\ln(3)}{12\ln(1.005)}, \infty\right)\)
36. \((-\infty, -1) \cup (0, 1)\)
37. \(\left(-\infty, \frac{\ln\left(\frac{2}{5}\right)}{\ln\left(\frac{4}{5}\right)} \right] = \left(-\infty, \frac{\ln(2)-\ln(5)}{\ln(4)-\ln(5)} \right]\)
38. \(\left(-\infty, \frac{\ln\left(\frac{2}{377}\right)}{-0.8} \right] = \left(-\infty, \frac{5}{4}\ln\left(\frac{377}{2}\right) \right]\)
39. \(\left[\frac{\ln\left(\frac{1}{18}\right)}{-0.1}, \infty\right) = [10\ln(18), \infty)\)
40. \(x \approx -0.76666, \, x = 2, \, x = 4\)
41. \(x \approx 0.01866, \, x \approx 1.7115\)
42. \(x = 0\)
43. \((-\infty, 1]\)
44. \(\approx (-\infty, 2.7095)\)
45. \(\approx (2.3217, 4.3717)\)
46. \(x > \frac{1}{3}(\ln(6) + 1)\)
47. \(f^{-1} = \ln\left(x + \sqrt{x^{2} + 1}\right)\). Both \(f\) and \(f^{-1}\) have domain \((-\infty, \infty)\) and range \((-\infty, \infty)\).