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# General Math Lesson 3

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# General Math Lesson 3

Exponential Expression
- Evaluating Exponential Expression
-Exponential Function, Equation and Inequality
- Solving Exponential Equation
- Solving Exponential Inequalitie

Exponential Expression
- Evaluating Exponential Expression
-Exponential Function, Equation and Inequality
- Solving Exponential Equation
- Solving Exponential Inequalitie

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### General Math Lesson 3

1. 1. 2
2. 2. 3
3. 3. 4
4. 4. 5 x -3 -2 -1 0 1 2 3 𝑦 = 1 3 𝑥 𝑦 = 10 𝑥 𝑦 = 0.8 𝑥 Note: if the exponent is a negative number always get the reciprocal of the base and change the sign of the given exponent. EXAMPLE 1: Complete the table of values for x = -3, -2, -1, 0, 1, 2, 3 for the exponential functions 𝑦 = 1 3 𝑥 , 𝑦 = 10 𝑥 𝑎𝑛𝑑 𝑦 = 0.8 𝑥
5. 5. 6 Solution:
6. 6. 7 x -3 -2 -1 0 1 2 3 𝑦 = 1 3 𝑥 27 9 3 1 𝟏 𝟑 𝟏 𝟗 𝟏 𝟐𝟕 𝑦 = 10 𝑥 𝑦 = 0.8 𝑥
7. 7. 8 Solution:
8. 8. 9 x -3 -2 -1 0 1 2 3 𝑦 = 1 3 𝑥 27 9 3 1 𝟏 𝟑 𝟏 𝟗 𝟏 𝟐𝟕 𝑦 = 10 𝑥 𝟏 1000 𝟏 100 𝟏 10 1 10 100 1000 𝑦 = 0.8 𝑥
9. 9. 10 Solution:
10. 10. 11 x -3 -2 -1 0 1 2 3 𝑦 = 1 3 𝑥 27 9 3 1 𝟏 𝟑 𝟏 𝟗 𝟏 𝟐𝟕 𝑦 = 10 𝑥 𝟏 1000 𝟏 100 𝟏 10 1 10 100 1000 𝑦 = 0.8 𝑥 1.953 1.563 1.25 1 0.8 0.64 0.512
11. 11. 12 EXPONENTIAL FUNCTION, INEQUALITIES AND EQUATION
12. 12. 13
13. 13. 14
14. 14. 15
15. 15. 16 𝑵𝑶𝑵𝑬 𝑶𝑭 𝑻𝑯𝑬𝑺𝑬 𝑬𝑿𝑷𝑶𝑵𝑬𝑵𝑻𝑰𝑨𝑳 𝑰𝑵𝑬𝑸𝑼𝑨𝑳𝑰𝑻𝒀 𝑬𝑿𝑷𝑶𝑵𝑬𝑵𝑻𝑰𝑨𝑳 𝑭𝑼𝑵𝑪𝑻𝑰𝑶𝑵
16. 16. 17 𝑵𝑶𝑵𝑬 𝑶𝑭 𝑻𝑯𝑬𝑺𝑬 𝑬𝑿𝑷𝑶𝑵𝑬𝑵𝑻𝑰𝑨𝑳 𝑬𝑸𝑼𝑨𝑻𝑰𝑶𝑵 𝑵𝑶𝑵𝑬 𝑶𝑭 𝑻𝑯𝑬𝑺𝑬
17. 17. 18
18. 18. 19
19. 19. 20
20. 20. 21 x -3 -2 -1 0 1 2 3 𝑦 = 1 5 𝑥 𝑦 = 152𝑥 𝑦 = 0.5 𝑥−1 Note: if the exponent is a negative number always get the reciprocal of the base and change the sign of the given exponent. Complete the table of values for x = -3, -2, -1, 0, 1, 2, 3 for the exponential functions 𝑦 = 1 5 𝑥 , 𝑦 = 152𝑥 𝑎𝑛𝑑 𝑦 = 0.5 𝑥−1
21. 21. 22
22. 22. 23 Exponential Equation Noneof these None of these Exponential function Exponential Equation
23. 23. 24 Noneof these Exponential function None of these Exponential inequalities Exponential equation
24. 24. 25 x -3 -2 -1 0 1 2 3 𝑦 = 1 5 𝑥 𝑦 = 152𝑥 𝑦 = 0.5 𝑥−1 Note: if the exponent is a negative number always get the reciprocal of the base and change the sign of the given exponent. Complete the table of values for x = -3, -2, -1, 0, 1, 2, 3 for the exponential functions 𝑦 = 1 5 𝑥 , 𝑦 = 152𝑥 𝑎𝑛𝑑 𝑦 = 0.5 𝑥−1
25. 25. 26 Solution:
26. 26. 27 Solution:
27. 27. 28 Solution:
28. 28. 29
29. 29. 30
30. 30. 31 x -3 -2 -1 0 1 2 3 𝑦 = 1 4 4𝑥−2 𝑦 = 1.53𝑥 𝑦 = 0.25 𝑥−3
31. 31. 32
32. 32. 33
33. 33. 34 𝑰𝒇 𝒙 𝟏 ≠ 𝒙 𝟐, 𝒕𝒉𝒆𝒏 𝒃 𝒙 𝟏 ≠ 𝒃 𝒙 𝟐. 𝑪𝒐𝒏𝒗𝒆𝒓𝒔𝒆𝒍𝒚, 𝒊𝒇 𝒙 𝟏 = 𝒙 𝟐, 𝒕𝒉𝒆𝒏 𝒃 𝒙 𝟏 = 𝒃 𝒙 𝟐.
34. 34. 35 Step 1: Determine if the numbers can be written using the same base. If so, go to Step 2. If not, stop and use Steps for Solving an Exponential Equation with Different Bases. Step 2: Rewrite the problem using the same base.
35. 35. 36 Step 3: Use the properties of exponents to simplify the problem. Step 4: Once the bases are the same, drop the bases and set the exponents equal to each other. Step 5: Finish solving the problem by isolating the variable.
36. 36. 37 EXAMPLE: Solve the Exponential Equation. 𝟐 𝒙 = 𝟒 𝟐 𝒙 = 𝟐 𝟐 𝒙 = 𝟐 𝟐 𝒙 = 𝟏 𝟐 𝟐 𝒙 = 𝟐−𝟏 𝒙 = −𝟏
37. 37. 38 𝟒 𝒙−𝟏 = 𝟏𝟔 EXAMPLE: Solve the Exponential Equation. 𝟒 𝒙−𝟏 = 𝟒 𝟐 𝒙 − 𝟏 = 𝟐 𝒙 = 𝟐 + 𝟏 𝒙 = 𝟑
38. 38. 39 EXAMPLE: Solve the Exponential Equation. 𝟏𝟐𝟓 𝒙−𝟏 = 𝟐𝟓 𝒙+𝟑 𝟓 𝟑 𝒙−𝟏 = 𝟓 𝟐 𝒙+𝟑 𝟑 𝒙 − 𝟏 = 𝟐 𝒙 + 𝟑 𝟑𝒙 − 𝟑 = 𝟐𝒙 + 𝟔 𝟑𝒙 − 𝟐𝒙 = 𝟔 + 𝟑 𝒙 = 𝟗
39. 39. 40 EXAMPLE: Solve the Exponential Equation. 𝟐 𝟑𝒙 = 𝟏𝟔 𝟏−𝒙 𝟐 𝟑𝒙 = 𝟐 𝟒 𝟏−𝒙 𝟑𝒙 = 𝟒 𝟏 − 𝒙 𝟑𝒙 = 𝟒 − 𝟒𝒙 𝟑𝒙 + 𝟒𝒙 = 𝟒 𝟕𝒙 = 𝟒 𝒙 = 𝟒 𝟕
40. 40. 41 EXAMPLE: Solve the Exponential Equation. 𝟗 𝒙 𝟐 = 𝟑 𝒙+𝟑 𝟑 𝟐 𝒙 𝟐 = 𝟑 𝒙+𝟑 𝟐𝒙 𝟐 = 𝒙 + 𝟑 𝟐𝒙 𝟐 − 𝒙 − 𝟑 = 𝟎 𝟐𝒙 − 𝟑 𝒙 + 𝟏 = 𝟎; 𝟐𝒙 − 𝟑 = 𝟎 𝒙 = 𝟑 𝟐 𝒙 + 𝟏 = 𝟎 𝒙 = −𝟏
41. 41. 42
42. 42. 43 If b > 1, then the exponential function 𝑦 = 𝑏 𝑥 is increasing for all x. this means that 𝑏 𝑥 < 𝑏 𝑦 if and only if x > y. Note: If the base is greater than one, the direction of the inequality is retained.
43. 43. 44 If 𝟎 < 𝒃 > 𝟏, then the exponential function 𝒚 = 𝒃 𝒙 is decreasing for all x. This means that 𝒃 𝒙 > 𝒃 𝒚 if and only if x < y. Note: If the base is less than one, the direction of the inequality is reversed.
44. 44. 45  If the same real number is added or subtracted from both sides of an inequality, the sense of the inequality is not changed.  If both sides of an inequality are multiplied by or divided by the same positive real number, the sense of the inequality is not changed.  If both sides of an inequality are multiplied by or divided by the same negative real number, the sense of the inequality is changed.
45. 45. 46 EXAMPLE: Solve the following Exponential Inequalities. 𝟑 𝒙+𝟏 > 𝟖𝟏 𝟑 𝒙+𝟏 > 𝟑 𝟒 𝒙 + 𝟏 > 𝟒 𝒙 > 𝟒 − 𝟏 𝒙 > 𝟑 Since the base is 3 > 1, the direction of the inequality is retained. Thus, the solution set is 𝟑, +∞
46. 46. 47 EXAMPLE: Solve the following Exponential Inequalities. 𝟐 𝟒𝒙+𝟏 ≤ 𝟓𝟏𝟐 𝟐 𝟒𝒙+𝟏 ≤ 𝟐 𝟗 𝟒𝒙 + 𝟏 ≤ 𝟗 𝟒𝒙 ≤ 𝟗 − 𝟏 𝟒𝒙 ≤ 𝟖 𝒙 ≤ 𝟐 Since the base is 2 > 1, the direction of the inequality is retained. Thus, the solution set is −∞, 𝟐
47. 47. 48 EXAMPLE: Solve the following Exponential Inequalities. 𝟑 𝒙 < 𝟗 𝒙−𝟐 𝟑 𝒙 < 𝟑 𝟐 𝒙−𝟐 𝒙 < 𝟐 𝒙 − 𝟐 𝒙 < 𝟐𝒙 − 𝟒 𝒙 − 𝟐𝒙 < −𝟒 −𝒙 < −𝟒 𝒙 > 𝟒 Since the base is 3 > 1, the direction of the inequality is retained. Thus, the solution set is 𝟒, +∞
48. 48. 49 EXAMPLE: Solve the following Exponential Inequalities. 𝟏 𝟏𝟎 𝒙+𝟓 ≥ 𝟏 𝟏𝟎𝟎 𝟑𝒙 𝟏 𝟏𝟎 𝒙+𝟓 ≥ 𝟏 𝟏𝟎 𝟐 𝟑𝒙 𝒙 + 𝟓 ≤ 𝟐 𝟑𝒙 𝒙 + 𝟓 ≤ 𝟔𝒙 𝒙 − 𝟔𝒙 ≤ −𝟓 −𝟓𝒙 ≤ −𝟓 𝒙 ≥ 𝟏 Since the base is 𝟏 𝟏𝟎 < 1, the direction of the inequality is reversed. Thus, the solution set is 𝟏, +∞
49. 49. 50
50. 50. 51 Solve the following exponential equation.
51. 51. 52 Solve the following exponential inequalities
52. 52. 53
53. 53. 54