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Questions and Answers
What is exponential growth?
What is exponential growth?
- A type of decay where the population decreases rapidly
- A fixed increase of a population over time
- Growth pattern that varies with time
- Growth pattern in which individuals in a population reproduce at a constant rate (correct)
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?
What is the basic form of exponential functions?
What is the basic form of exponential functions?
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.
What are exponential inequalities?
What are exponential inequalities?
What does the logarithm base b of a number x represent?
What does the logarithm base b of a number x represent?
Logarithmic functions behave differently than other parent functions.
Logarithmic functions behave differently than other parent functions.
What is the equality property for logarithmic equations?
What is the equality property for logarithmic equations?
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
- 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
- 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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