Exponential Functions — SAT Math Guide
Exponential functions SAT questions test your ability to work with rapidly growing or decaying quantities. These functions appear in real-world contexts like population growth, compound interest, and radioactive decay. You'll encounter 2-3 exponential function problems in the SAT math section's Advanced Math domain. With the right approach, these questions become straightforward points toward your target score.
What You Need to Know
Exponential functions have the form f(x) = a · b^x where a ≠ 0 and b > 0, b ≠ 1
The base b determines growth (b > 1) or decay (0 < b < 1)
Growth factor = 1 + rate, decay factor = 1 - rate
Compound interest formula: A = P(1 + r)^t
Half-life and doubling time create specific exponential relationships
Exponential equations often require logarithms or substitution to solve
📐 KEY FORMULA: f(x) = a · b^x (standard form) or f(x) = a · e^(kx) (natural exponential)
💡 PRO TIP: When solving exponential equations, look for opportunities to express both sides with the same base.
How to Solve Exponential Functions on the SAT
Example Question 1 — Medium Difficulty
A population of bacteria doubles every 3 hours. If there are initially 500 bacteria, which function represents the population P after t hours?
A) P(t) = 500 · 2^t
B) P(t) = 500 · 2^(t/3)
C) P(t) = 500 · 3^(t/2)
D) P(t) = 1000 · 2^t
Solution:
Step 1: Identify that doubling every 3 hours means the base should be 2, but the exponent needs adjustment.
Step 2: Since doubling occurs every 3 hours, after t hours there are t/3 doubling periods.
Step 3: The function becomes P(t) = 500 · 2^(t/3).
✅Answer: B — The exponent t/3 correctly represents the number of 3-hour periods in t hours.
Example Question 2 — Hard Difficulty
The value of a car depreciates according to the function V(t) = 25000(0.85)^t, where t is the number of years since purchase. After how many complete years will the car's value first drop below $10,000?
A) 8 years
B) 9 years
C) 10 years
D) 11 years
Solution:
Step 1: Set up the inequality 25000(0.85)^t < 10000.
Step 2: Divide both sides by 25000: (0.85)^t < 0.4.
Step 3: Use logarithms or test values. Testing: (0.85)^10 ≈ 0.197 and (0.85)^9 ≈ 0.232.
Step 4: Since 0.232 > 0.4 is false but 0.197 < 0.4 is true, the value drops below $10,000 after 10 years.
✅Answer: C — After 10 complete years, the car's value first drops below the $10,000 threshold.
Common SAT Math Mistakes to Avoid
❌Mistake: Confusing growth rate with growth factor (using 0.05 instead of 1.05 for 5% growth)
✅Fix: Remember growth factor = 1 + rate, decay factor = 1 - rate
❌Mistake: Mishandling time periods in compound growth (using t instead of t/n for periods)
✅Fix: Carefully identify how often the change occurs and adjust your exponent accordingly
❌Mistake: Forgetting to check if your exponential solution makes sense in context
✅Fix: Always verify that growth/decay matches the problem's real-world scenario
❌Mistake: Mixing up exponential and linear growth patterns
✅Fix: Exponential means constant percentage change, linear means constant amount change
Practice Question — Try It Yourself
The amount of a radioactive substance remaining after t years is given by A(t) = 100e^(-0.0693t). What is the half-life of this substance?
A) 5 years
B) 10 years
C) 15 years
D) 20 years
Show Answer
Answer: B — At half-life, A(t) = 50. Setting 100e^(-0.0693t) = 50 gives e^(-0.0693t) = 0.5. Taking the natural logarithm: -0.0693t = ln(0.5) ≈ -0.693, so t = 10 years.
Key Takeaways for the SAT
Master the relationship between growth rates and growth factors for Digital SAT success
Practice identifying whether time periods match the growth/decay interval
SAT math exponential functions often involve compound interest, population models, or decay scenarios
Use substitution or logarithms strategically when bases can't be matched easily
Always check that your mathematical solution makes sense in the problem's context
Related SAT Math Topics
Strengthen your SAT math prep with these related topics:
Quadratic functions →
Systems of equations →