Pre-Calculus Unit 3: Exponential and Logarithmic Functions — Free Review Games.
This unit covers exponential models, logarithmic properties, natural log and solving log equations — essential concepts for Pre-Calculus. Use our interactive study games to test your understanding, or review questions in traditional format below.
Pick a mode. Play.
Answer questions as fast as you can. 2 minutes on the clock. Build streaks for bonus points!
Don't want to play?
Review the questions traditionally. Click to expand.
Questions loading...
Focus on understanding.
Focus on understanding core concepts before memorizing details. Use the game modes to test yourself repeatedly — spaced repetition is proven to boost long-term retention.
Related units
This unit covers exponential models, logarithmic properties, natural log and solving log equations — essential concepts for Pre-Calculus. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Exponential Models
Exponential functions have the form f(x) = a·b^x, where a is the initial value and b is the base (growth factor). If b > 1 the function models growth; if 0 < b < 1 it models decay. Students must identify these components from tables, graphs, and word problems.
Key Points
- Growth: b > 1; Decay: 0 < b < 1
- The y-intercept is always the initial value a (when x = 0)
- Doubling/halving time problems use the form A(t) = A₀·b^(t/period)
- Percent growth rate r converts to base: b = 1 + r (growth) or b = 1 − r (decay)
A population of 500 bacteria doubles every 3 hours. Write a function for population P after t hours, then find the population after 9 hours.
The initial value is 500 and the doubling period is 3, so P(t) = 500·2^(t/3). At t = 9, P(9) = 500·2^(9/3) = 500·2^3 = 500·8 = 4000. The exponent t/3 counts how many doubling periods have passed.
2 Logarithmic Properties
Logarithms are the inverse of exponential functions: log_b(x) = y means b^y = x. The three core properties — product, quotient, and power rules — allow students to expand, condense, and simplify logarithmic expressions. These properties appear in nearly every log equation on the exam.
Key Points
- Product rule: log_b(MN) = log_b(M) + log_b(N)
- Quotient rule: log_b(M/N) = log_b(M) − log_b(N)
- Power rule: log_b(M^p) = p·log_b(M)
- Change of base: log_b(x) = log(x)/log(b) = ln(x)/ln(b)
Expand completely: log₂(8x³/y)
Apply the quotient rule first: log₂(8x³) − log₂(y). Then apply the product rule to the numerator: log₂(8) + log₂(x³) − log₂(y). Finally apply the power rule and evaluate log₂(8) = 3, giving 3 + 3·log₂(x) − log₂(y).
3 Natural Log
The natural logarithm ln(x) is log base e, where e ≈ 2.718. It is the inverse of e^x, so ln(e^x) = x and e^(ln x) = x. Students must recognize and use these inverse relationships to solve equations involving e.
Key Points
- ln(e) = 1 and ln(1) = 0
- ln(e^x) = x and e^(ln x) = x (inverse cancellation)
- All logarithmic properties apply to ln exactly as they do to log
- Continuous growth/decay uses A(t) = A₀·e^(rt); r > 0 is growth, r < 0 is decay
Solve: e^(2x−1) = 5
Take the natural log of both sides: ln(e^(2x−1)) = ln(5). The left side simplifies by cancellation to 2x − 1 = ln(5). Adding 1 and dividing by 2 gives x = (1 + ln 5)/2 ≈ (1 + 1.609)/2 ≈ 1.305.
4 Solving Log Equations
Log equations are solved by either converting to exponential form (if there is one log term) or using properties to condense both sides to a single log before converting. Students must always check for extraneous solutions by verifying that the argument of every logarithm is positive in the original equation.
Key Points
- One log term: isolate it, then rewrite as b^y = x
- Multiple log terms: condense using properties, then convert to exponential form
- Always check: the argument of a log must be > 0 (reject extraneous solutions)
- If log_b(A) = log_b(B), then A = B (one-to-one property)
Solve: log₃(x + 6) + log₃(x) = 3
Use the product rule to condense the left side: log₃(x(x + 6)) = 3. Convert to exponential form: x(x + 6) = 3³ = 27, which gives x² + 6x − 27 = 0 and factors to (x + 9)(x − 3) = 0. x = −9 is extraneous because log₃(−9) is undefined, so the only solution is x = 3.
Questions, answered.
What is Exponential and Logarithmic Functions?
Exponential and Logarithmic Functions is Unit 3 of Pre-Calculus, covering exponential models, logarithmic properties, natural log and solving log equations.
How to study for Pre-Calculus Unit 3?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 28+ review questions across 5 different game modes.