Exponential Functions: Theory, Applications, and
Practice Problems
Complete Study Guide with Step-by-Step Solution
Prepared by: Anggara Prima Sektiaji
2025
,A. Summary of Exponents
1. General form: 𝑎𝑛 , where:
𝑎 = base
𝑛 = exponent
Basic rules:
• 𝑎𝑚 ⋅ 𝑎𝑛 = 𝑎𝑚+𝑛
𝑎𝑚
• = 𝑎𝑚−𝑛 for 𝑎 ≠ 0
𝑎𝑛
• (𝑎𝑚 )𝑛 = 𝑎𝑚.𝑛
• 𝑎0 = 1 for 𝑎 ≠ 0
1
• 𝑎−𝑛 = 𝑎𝑛
𝑚
•
𝑛
𝑎 𝑛 = √𝑎 𝑚
2. Exponential Functions
General form: 𝑦 = 𝑎 𝑥 with 𝑎 > 0 and 𝑎 ≠ 1
• If 𝑎 > 1 → increasing (exponential growth).
• If 0 < 𝑎 < 10 → decreasing (exponential decay).
3. Applications of Exponents
• Population growth: 𝑃(𝑡) = 𝑃0 (1 + 𝑟)𝑡
𝑃(𝑡) = population size after time 𝑡
𝑃0 = initial population (when 𝑡 = 0)
𝑟 = growth rate (as a decimal; e.g., 5% = 0.05)
𝑡 = time (measured in year)
• Radioactive decay/Bacteria: 𝑁(𝑡) = 𝑁0 ∙ 𝑘 𝑡
𝑁(𝑡) = the amount after ttt periods
𝑁0 = the initial amount
𝑘 = the growth/decay factor per period
If 𝑘 > 1 → growth
If 0 < 𝑘 < 1→ decay (reduction)
𝑡 = the number of periods (for example, hours, days, or years)
, B. Practice Problems – Exponents
Easy
1. Compute 23 . 24
56
2. Simplify 52
3. Find the value of 100
1
4. Express 32 using a negative exponent
5. Compute (23 )2
Medium
3
6. Simplify 92
7. If 2𝑥 = 32, find the value of 𝑥
2
8. Simplify 83
2
9. Find the simplified form of 273
10. If 𝑎 𝑥+2 = 𝑎5 , determine the value of 𝑥
Expert
11. Solve 32𝑥−1 = 81
12. If 52𝑥 = 125, determine 𝑥
13. Solve for 𝑥: 2𝑥 ⋅ 4𝑥−1 = 32
14. The graph of 𝑦 = 2𝑥 intersects the line 𝑦 = 8. Find the coordinates of the intersection
point
15. A population of bacteria doubles every 3 hours. If initially there are 500 bacteria, how
many will there be after 12 hours?
16. Solve 7𝑥+1 = 1
17. If 𝑎3𝑥 = 0,125 and 𝑎 = 2 , find 𝑥
18. An investment of $1,000 grows to $2,500 in 10 years with annual compound interest.
Find the interest rate 𝑟
3𝑥
19. Solve: 9𝑥−1 = 27
1 𝑥 1
20. The graph of 𝑦 = (2) intersects the line 𝑦 = 16 . Find the value of 𝑥
Practice Problems
Complete Study Guide with Step-by-Step Solution
Prepared by: Anggara Prima Sektiaji
2025
,A. Summary of Exponents
1. General form: 𝑎𝑛 , where:
𝑎 = base
𝑛 = exponent
Basic rules:
• 𝑎𝑚 ⋅ 𝑎𝑛 = 𝑎𝑚+𝑛
𝑎𝑚
• = 𝑎𝑚−𝑛 for 𝑎 ≠ 0
𝑎𝑛
• (𝑎𝑚 )𝑛 = 𝑎𝑚.𝑛
• 𝑎0 = 1 for 𝑎 ≠ 0
1
• 𝑎−𝑛 = 𝑎𝑛
𝑚
•
𝑛
𝑎 𝑛 = √𝑎 𝑚
2. Exponential Functions
General form: 𝑦 = 𝑎 𝑥 with 𝑎 > 0 and 𝑎 ≠ 1
• If 𝑎 > 1 → increasing (exponential growth).
• If 0 < 𝑎 < 10 → decreasing (exponential decay).
3. Applications of Exponents
• Population growth: 𝑃(𝑡) = 𝑃0 (1 + 𝑟)𝑡
𝑃(𝑡) = population size after time 𝑡
𝑃0 = initial population (when 𝑡 = 0)
𝑟 = growth rate (as a decimal; e.g., 5% = 0.05)
𝑡 = time (measured in year)
• Radioactive decay/Bacteria: 𝑁(𝑡) = 𝑁0 ∙ 𝑘 𝑡
𝑁(𝑡) = the amount after ttt periods
𝑁0 = the initial amount
𝑘 = the growth/decay factor per period
If 𝑘 > 1 → growth
If 0 < 𝑘 < 1→ decay (reduction)
𝑡 = the number of periods (for example, hours, days, or years)
, B. Practice Problems – Exponents
Easy
1. Compute 23 . 24
56
2. Simplify 52
3. Find the value of 100
1
4. Express 32 using a negative exponent
5. Compute (23 )2
Medium
3
6. Simplify 92
7. If 2𝑥 = 32, find the value of 𝑥
2
8. Simplify 83
2
9. Find the simplified form of 273
10. If 𝑎 𝑥+2 = 𝑎5 , determine the value of 𝑥
Expert
11. Solve 32𝑥−1 = 81
12. If 52𝑥 = 125, determine 𝑥
13. Solve for 𝑥: 2𝑥 ⋅ 4𝑥−1 = 32
14. The graph of 𝑦 = 2𝑥 intersects the line 𝑦 = 8. Find the coordinates of the intersection
point
15. A population of bacteria doubles every 3 hours. If initially there are 500 bacteria, how
many will there be after 12 hours?
16. Solve 7𝑥+1 = 1
17. If 𝑎3𝑥 = 0,125 and 𝑎 = 2 , find 𝑥
18. An investment of $1,000 grows to $2,500 in 10 years with annual compound interest.
Find the interest rate 𝑟
3𝑥
19. Solve: 9𝑥−1 = 27
1 𝑥 1
20. The graph of 𝑦 = (2) intersects the line 𝑦 = 16 . Find the value of 𝑥
