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Algebra II terms - Exponentiation (Chapter 6)

Terms in this set (23)

A function whose general equation is
𝑦 = 𝑎 • 𝘣ˣ
where 𝑎 and 𝘣 stand for constants, 𝘣 is positive, and 𝑥 and 𝑦 are independent and dependent variables, respectively.
EXPONENTIAL FUNCTION
𝑥ⁿ means the product of 𝑛 𝑥's.
EXPONENTIATION FOR POSITIVE INTEGER EXPONENTS
NAMES:
In the expression 𝑥ⁿ,

1. 𝑥 is called the__________,
2. 𝑛 is called the __________________,
3. 𝑥ⁿ (the whole thing) is called a _____________.
base, exponent, power
𝑥ª • 𝑥ᵇ = 𝑥ª⁺ᵇ
Product of two powers (with equal bases)
𝑥ᵃ/𝑥ᵇ = 𝑥ᵃ⁻ᵇ
Quotient of two powers (with equal bases)
(𝑥ᵃ)ᵇ = 𝑥ᵃᵇ
Power of a power
(𝑥𝑦)ᵃ = 𝑥ᵃ𝑦ᵃ
Power of a product
(𝑥/𝑦)ᵃ = 𝑥ᵃ/𝑦ᵃ
Power of a quotient
𝑥⁻ⁿ = 1/𝑥ⁿ
Exponentiation for negative exponents
𝑥⁰ = 1
Exponentiation for 0 exponent
𝑥^ª/ᵇ = (ᵇ√𝑥)ª
Exponentiation for fractional exponents
𝑥^¹/𝘯 = ⁿ√𝑥
Exponentiation for reciprocal exponents
A number is in this form if and only if it is written in the form

mantissa ╳ 10^characteristic

where the mantissa is a number between 1 and 10 (including 1, but not including 10), and the characteristic is an integer that tells how many places the decimal point has been moved.
Scientific notation
y = log 𝑥 if and only if 10^𝑦 = 𝑥
Base 10 logarithm
log 10ˣ = 𝑥
Logarithm of a power of 10
𝑦 = logᵦ 𝑥 if and only if 𝑏ʸ = 𝑥
Base b logarithm
logᵦ (𝑥𝑦) = logᵦ 𝑥 + logᵦ 𝑦
Logarithm of a product
logᵦ (ˣ/ᵧ) = logᵦ 𝑥 — logᵦ 𝑦
Logarithm of a quotient
logᵦ (𝑥ⁿ) = 𝑛 logᵦ 𝑥
Logarithm of a power
logᵦ 𝑥 = (ˡᵒᵍ𝘢 ˣ/ʟᴏɢₐ 𝘣)
Change of base property of logarithms
If 𝒇 is a function, then the inverse of 𝒇, abbreviated 𝒇⁻¹ and pronounced "𝒇-inverse," is the relation obtained by interchanging the two variables.
Inverse of a function
If 𝒇(𝑥) = 𝑎 ╳ 𝘣ˣ, then 𝒇(𝑥 + 𝑐) = 𝒇(𝑥) ╳ 𝘣ᶜ.

(Adding a constant to the value of 𝑥 multiplies 𝒇(𝑥) by a (different) constant.)
Add-multiply property of exponential functions
If 𝒇(𝑥) = 𝑚𝑥 + 𝘣, then 𝒇(𝑥 + 𝑐) = 𝒇(𝑥) + 𝑚𝑐.

( Adding a constant to the value of 𝑥 adds a (different) constant to the value of 𝒇(𝑥). )
Add-add property of linear functions
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