Questions and Answers
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?
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?
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?
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What do the properties of logarithms state about the quotient?
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What do the properties of logarithms state about power?
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What are negative exponents?
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What is the form of a logarithmic function?
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What is the simple interest formula?
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What is the compound interest formula?
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What is the relationship between logarithmic functions and exponential functions?
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How can properties of logarithms be used to simplify equations?
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When is the base change property used?
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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.
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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.