Algebra 2 Module 7 Flashcards

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.