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Questions and Answers
What is exponential growth?
What is exponential growth?
Define exponential decay.
Define exponential decay.
When an initial amount decreases by the same percent over a given period of time.
The growth rate is the ______ by which a quantity increases (or decreases) over time.
The growth rate is the ______ by which a quantity increases (or decreases) over time.
addend
What is the decay factor?
What is the decay factor?
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What is the basic form of exponential functions?
What is the basic form of exponential functions?
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If b is a positive number other than 1, then b^x = b^y if and only if x = y.
If b is a positive number other than 1, then b^x = b^y if and only if x = y.
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What are exponential inequalities?
What are exponential inequalities?
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What does the logarithm base b of a number x represent?
What does the logarithm base b of a number x represent?
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Logarithmic functions behave differently than other parent functions.
Logarithmic functions behave differently than other parent functions.
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What is the equality property for logarithmic equations?
What is the equality property for logarithmic equations?
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In logarithmic functions, if a is greater than 1, then if x > y, it implies x < y.
In logarithmic functions, if a is greater than 1, then if x > y, it implies x < y.
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Study Notes
Exponential Growth and Decay
- Exponential Growth: Individuals in a population reproduce at a constant rate, leading to rapid increase.
- Exponential Decay: An initial amount decreases consistently by a fixed percentage over time.
Growth and Decay Factors
- Growth Factor: The addend that represents the rate of increase or decrease; a 10% annual investment yield is an example where the total multiplies by 1.10 (110%).
- Decay Factor: Calculated as 1 minus the percentage change expressed as a decimal in decay scenarios.
Transformations of Functions
- Exponential Functions Transformation: The base function f(x) = b^x can be altered through vertical/horizontal shifts and graph reversals, modifying its appearance but not its fundamental characteristics.
- Logarithmic Functions Transformation: Similar transformations apply to logarithmic graphs, allowing shifts, stretches, compressions, and reflections without altering shape.
Properties of Exponential Functions
- Property of Equality: For a positive base b (b ≠ 1), if b^x = b^y, then x must equal y; for instance, if 2^x = 2^8, then x = 8.
- Property of Inequality: Exponential inequalities involve variable exponents and are essential for comparing quantities through repeated multiplication.
Properties of Logarithms
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Logarithmic Properties: The logarithm base b of a number x indicates the power to which b must be raised to reach x. Key properties include:
- Product Rule: The log of a product equals the sum of the logs of the factors.
Properties of Equality and Inequality in Logarithms
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Equality Properties:
- For exponential equations, if b is a positive number (b ≠ 1), bx = by implies x = y by matching bases and equating exponents.
- For logarithmic equations, log_b(x) = log_b(y) holds true when x = y, with b, x, and y being positive numbers (b ≠ 1).
- Inequality Properties: If a > 1, then inequalities involving logarithmic expressions follow standard comparison rules (x > y or x < y).
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Description
Test your understanding of exponents and logarithms with this flashcard quiz. Learn definitions and key concepts such as exponential growth, decay, and growth factors. Perfect for Algebra 2 students looking to reinforce their knowledge.
