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Questions and Answers
What is an exponential function?
What is an exponential function?
An exponential function is a function where the variable is an exponent and the base is a positive number not equal to 1, of the form f(x) = b^x where b > 0.
What is an exponential growth function?
What is an exponential growth function?
A function where the exponent is a variable and the base is positive, resulting in an increase over time, such as f(x) = 10^x.
What is an exponential decay function?
What is an exponential decay function?
A function where the base is a number between 0 and 1, leading to a decrease over time.
What do the properties of logarithms state about the product?
What do the properties of logarithms state about the product?
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What do the properties of logarithms state about the quotient?
What do the properties of logarithms state about the quotient?
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What do the properties of logarithms state about power?
What do the properties of logarithms state about power?
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What are negative exponents?
What are negative exponents?
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What is the form of a logarithmic function?
What is the form of a logarithmic function?
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What is the simple interest formula?
What is the simple interest formula?
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What is the compound interest formula?
What is the compound interest formula?
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What is the relationship between logarithmic functions and exponential functions?
What is the relationship between logarithmic functions and exponential functions?
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How can properties of logarithms be used to simplify equations?
How can properties of logarithms be used to simplify equations?
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When is the base change property used?
When is the base change property used?
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How do you graph a log function?
How do you graph a log function?
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Study Notes
Exponential Functions
- An exponential function has the variable as the exponent with the base being a positive number not equal to 1, expressed as f(x) = b^x.
- Example: f(x) = 3^x is an exponential function, whereas f(x) = x^2 is not.
Exponential Growth Function
- Characterized by a variable exponent with a positive base, representing situations that increase over time.
- Common example: Compound interest in savings, modeled by f(x) = 10^x.
Exponential Decay Function
- Involves a base between 0 and 1, indicating a decrease over time.
- Example: Diminishing population of an endangered species, represented graphically over time.
Logarithm Properties
- Product Rule: Convert multiplication into logs by adding: logb(mn) = logb(m) + logb(n).
- Quotient Rule: Convert division into logs by subtracting: logb(m/n) = logb(m) - logb(n).
- Power Rule: Turn a power into multiplication: logb(m^k) = k * logb(m).
Exponential Properties
- Product Rule: Sum the exponents when multiplying like bases: d² × d³ = d⁵.
- Quotient Rule: Subtract exponents when dividing like bases: d⁵ ÷ d² = d³.
- Power Rule: Multiply exponents when raising a power to another power: (d³)³ = d⁹.
Negative Exponents
- Defined as the reciprocal of the base: 5⁻³ = 1/5³, simplifying to 0.008.
Logarithmic Function
- Represented as y = logb(x), with y being the logarithmic form and b^y = x as the exponential form.
Interest Formulas
- Simple Interest Formula: A = P(1 + rt) where P is principal, r is rate, and t is time.
- Compound Interest Formula: A = P(1 + r/n)^(nt), factoring in compounding periods.
Relationship Between Logarithmic and Exponential Functions
- Logarithmic functions serve as the inverses of exponential functions, reflecting their behavior graphically.
Simplifying Equations with Logarithm Properties
- Employ logarithmic properties to condense or decompose equations, enhancing their ease of solution.
Base Change Property
- Utilized when arguments of logarithms are not rational powers of the base: logb(a) can be changed using logx(a) / logx(b).
Graphing Log Functions
- Graphing involves inputting x-values to calculate corresponding y-values, then plotting the resulting coordinates.
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your knowledge with these flashcards focused on exponential functions and their properties. This quiz covers key concepts and definitions from Algebra 2 Module 7, making it an essential tool for mastering the subject.
