Algebra Properties

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.

$$a + b = 7 + 2 = 9$$

$$b + a = 2 + 7 = 9$$

Associativity Property of Algebra

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

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

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

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.

$$(a + b) \cdot c = a \cdot c + b \cdot c$$

$$a \cdot (b + c) = a \cdot b + a \cdot c$$

Additive Identity Property

$$a = a + 0$$

Multiplicative Identity Property

$$a = a \cdot 1 = 1 \cdot a$$

Additive Inverse Property

$$a + (-a) = 0 = (-a) + a$$

Multiplicative Inverse Property

$$2 \cdot \frac{1}{2} = 0 = \frac{1}{2} \cdot 2$$

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:

$$\log_b (xy) = \log_b x + \log_b y$$

Quotient Rule

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

$$\log_b (x/y) = \log_b x - \log_b y$$

Power Rule

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

$$\log_b (x^n) = n \log_b x$$

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:

$$\log_b x = \log_a x / \log_a b$$

Base Switch Rule

$$\log_b x = \frac{1}{\log_x b}$$