This document provides information and examples about exponential functions, equations, and inequalities. It discusses key properties of exponential functions including how the direction of an inequality changes depending on whether the base is greater or less than one. Steps are provided for solving exponential equations, which involve rewriting the problem so bases are the same, then isolating the variable. Examples demonstrate applying these steps to solve various exponential equations. Properties of inequalities are also discussed, such as how adding or multiplying the same term does not change the sense of an inequality.

2. 2

3. 3

4. 4

5. 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

6. 6
Solution:

7. 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

8. 8
Solution:

9. 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

10. 10
Solution:

11. 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

12. 12
EXPONENTIAL
FUNCTION,
INEQUALITIES AND
EQUATION

13. 13

14. 14

15. 15

16. 16
𝑵𝑶𝑵𝑬 𝑶𝑭 𝑻𝑯𝑬𝑺𝑬
𝑬𝑿𝑷𝑶𝑵𝑬𝑵𝑻𝑰𝑨𝑳 𝑰𝑵𝑬𝑸𝑼𝑨𝑳𝑰𝑻𝒀
𝑬𝑿𝑷𝑶𝑵𝑬𝑵𝑻𝑰𝑨𝑳 𝑭𝑼𝑵𝑪𝑻𝑰𝑶𝑵

17. 17
𝑵𝑶𝑵𝑬 𝑶𝑭 𝑻𝑯𝑬𝑺𝑬
𝑬𝑿𝑷𝑶𝑵𝑬𝑵𝑻𝑰𝑨𝑳 𝑬𝑸𝑼𝑨𝑻𝑰𝑶𝑵
𝑵𝑶𝑵𝑬 𝑶𝑭 𝑻𝑯𝑬𝑺𝑬

18. 18

19. 19

20. 20

21. 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

22. 22

23. 23
Exponential Equation
Noneof these
None of these
Exponential function
Exponential Equation

24. 24
Noneof these
Exponential function
None of these
Exponential inequalities
Exponential equation

25. 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

26. 26
Solution:

27. 27
Solution:

28. 28
Solution:

29. 29

30. 30

31. 31
x -3 -2 -1 0 1 2 3
𝑦 =
1
4
4𝑥−2
𝑦 = 1.53𝑥
𝑦 = 0.25 𝑥−3

32. 32

33. 33

34. 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.

35. 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.

36. 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.

37. 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

38. 38
𝟒 𝒙−𝟏 = 𝟏𝟔
EXAMPLE: Solve the Exponential Equation.
𝟒 𝒙−𝟏
= 𝟒 𝟐
𝒙 − 𝟏 = 𝟐
𝒙 = 𝟐 + 𝟏
𝒙 = 𝟑

39. 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

40. 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

41. 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

42. 42

43. 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.

44. 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.

45. 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.

46. 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, +∞

47. 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

48. 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, +∞

49. 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, +∞

50. 50

51. 51
Solve the following exponential equation.

52. 52
Solve the following exponential inequalities

53. 53

54. 54