Unit 7 of Algebra 2: Exponential and Logarithmic Functions.
This unit covers exponential growth and decay, logarithm properties and solving exponential equations — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers exponential growth and decay, logarithm properties and solving exponential equations — essential concepts for Algebra 2. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Exponential Growth and Decay
Students must know the standard form y = a·b^x, where a is the initial value and b is the base (b > 1 for growth, 0 < b < 1 for decay). They must also recognize and apply the continuous form y = a·e^(rt) and the half-life/doubling-time formula. Exams frequently require identifying growth vs. decay, finding initial values, and evaluating or solving the function at a given time.
Key Points
- Growth: b > 1; Decay: 0 < b < 1 in y = a·b^x
- Percent rate r converts to base b as: b = 1 + r (growth) or b = 1 − r (decay)
- Half-life formula: y = a·(1/2)^(t/h), where h is the half-life period
- The y-intercept is always a (the initial amount) when x = 0
A population of 500 bacteria doubles every 3 hours. Write an equation and find the population after 9 hours.
The initial value a = 500 and the doubling time is 3 hours, so the equation is y = 500·2^(t/3). Substituting t = 9 gives y = 500·2^(9/3) = 500·2^3 = 500·8 = 4000. The population after 9 hours is 4,000 bacteria.
2 Logarithm Properties
Students must know the three core logarithm properties — Product, Quotient, and Power Rules — and be able to apply them to expand or condense logarithmic expressions. They must also understand the change-of-base formula to evaluate non-standard bases on a calculator. Exams test both directions: expanding a single log into multiple terms and condensing multiple terms into one log.
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) or ln(x)/ln(b)
Condense into a single logarithm: 3·log(x) + log(y) − log(z)
Apply the Power Rule first to get log(x^3) + log(y) − log(z). Then apply the Product Rule to the addition: log(x^3·y) − log(z). Finally apply the Quotient Rule to get log(x^3·y / z). The condensed expression is log(x³y/z).
3 Solving Exponential Equations
Students must be able to solve exponential equations using two methods: rewriting both sides with a common base (when possible) and taking the logarithm of both sides (when bases cannot be matched). They must isolate the exponential expression before applying a logarithm and know when to use ln vs. log. Exams require exact answers in log form and approximate decimal answers.
Key Points
- If bases can match: set exponents equal and solve (e.g., 2^x = 8 → x = 3)
- If bases cannot match: take log or ln of both sides and use the Power Rule to bring down the exponent
- Isolate the exponential term before taking any logarithm
- Check for extraneous solutions when the equation involves sums or variable bases
Solve for x: 5·e^(2x) = 75
First isolate the exponential by dividing both sides by 5: e^(2x) = 15. Take the natural log of both sides: ln(e^(2x)) = ln(15), which simplifies to 2x = ln(15). Divide both sides by 2 to get x = ln(15)/2 ≈ 1.354.
Questions, answered.
What is Exponential and Logarithmic Functions?
Exponential and Logarithmic Functions is Unit 7 of Algebra 2, covering exponential growth and decay, logarithm properties and solving exponential equations.
How to study for Algebra 2 Unit 7?
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.