Den Dribbles

Algebra Properties

June 21, 2020

Resources

  1. Deriving the Quadratic Formula

Properties of Algebra

Commutative Property of Algebra

Definition:

  1. involving the condition that a group of quantities connected by operators gives the same result whatever the order of the quantities involved, e.g. a × b = b × a.
  2. relating to or involving substitution or exchange.
a+b=7+2=9a + b = 7 + 2 = 9
b+a=2+7=9b + a = 2 + 7 = 9

Associativity Property of Algebra

Grouping of more than two numbers to perform basic aritmetic operations of addition/multiplication does not affect the final result.

(a+b)+c=(2+4)+5=7(a + b) + c = (-2 + 4) + 5 = 7
a+(b+c)=2+(4+5)=7a + (b + c) = -2 + (4 + 5) = 7

Distributive Property of Algebra

The distributive property defines that the product of a single term and a sum or difference of two or more terms inside the bracket is same as multiplying each addend by the single term and then adding or subtracting the products.

(a+b)c=ac+bc(a + b) \cdot c = a \cdot c + b \cdot c
a(b+c)=ab+aca \cdot (b + c) = a \cdot b + a \cdot c

Additive Identity Property

a=a+0a = a + 0

Multiplicative Identity Property

a=a1=1aa = a \cdot 1 = 1 \cdot a

Additive Inverse Property

a+(a)=0=(a)+aa + (-a) = 0 = (-a) + a

Multiplicative Inverse Property

212=0=1222 \cdot \frac{1}{2} = 0 = \frac{1}{2} \cdot 2

Logarithmic Properties

Product Rule

The log of a product is equal to the sum of the log of the first base and the log of the second base:

logb(xy)=logbx+logby\log_b (xy) = \log_b x + \log_b y

Quotient Rule

The log of a quotient is equal to the difference of the logs of the numerator and denominator:

logb(x/y)=logbxlogby\log_b (x/y) = \log_b x - \log_b y

Power Rule

The log of a power is equal to the power times the log of the base:

logb(xn)=nlogbx\log_b (x^n) = n \log_b x

Change of Base Formula

The log of a new base is the log of the new base divided by the log of the old base in the new base:

logbx=logax/logab\log_b x = \log_a x / \log_a b

Base Switch Rule

logbx=1logxb\log_b x = \frac{1}{\log_x b}

A personal blog on all things of interest. Written by Dennis O'Keeffe, Follow me on Twitter