Algebra 2 Module 7 Flashcards

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?

For the product, you turn the numbers being multiplied into their own logarithms and add them together. Signup and view all the answers

What do the properties of logarithms state about the quotient?

For the quotient, turn the numbers being divided into their own logarithms and subtract them. Signup and view all the answers

What do the properties of logarithms state about power?

When a power is raised to another power, you multiply the exponents together. Signup and view all the answers

What are negative exponents?

Negative exponents are the reciprocal of their base. Signup and view all the answers

What is the form of a logarithmic function?

y = log_b(x), where b is the base and x is the argument. Signup and view all the answers

What is the simple interest formula?

A = P(1 + rt) Signup and view all the answers

What is the compound interest formula?

A = P (1 + r/n) ^ (nt) Signup and view all the answers

What is the relationship between logarithmic functions and exponential functions?

Logarithmic functions are the inverses of exponential functions. Signup and view all the answers

How can properties of logarithms be used to simplify equations?

You can use the properties of logarithms to condense or break down equations. Signup and view all the answers

When is the base change property used?

The base change formula is used when the argument is not a rational power of the base. Signup and view all the answers

How do you graph a log function?

Plug in values for x and solve for y, then graph the coordinates. Signup and view all the answers

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.

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.