Review Session #2

Outline Logarithms Solving Systems of Equations Factoring and Roots

Logarithmic Functions Consider the function x = b y (b > 0, b 1). Then the value y is called the logarithm of x with base b and is denoted by y = log b x. In other words, y = log b x if and only if x = b y. log b x = the power you would need to raise b to in order to get an answer of x. NOTE: x must be positive.

Exercise 1 Determine the value of the following terms: a) log 3 27 b) log 7 1 c) log 1/3 9 d) log Exercise 2 Solve for x in the following expressions: a) log 2 x = 5 b) log x 9 = 2

Laws of Logarithms

Exercise 3 Use the Laws of Logarithms to simplify the following expressions: a) log x 2 (x + 1) 4 b) Exercise 4 Use the Laws of Logarithms to solve for x in the following expressions: a) b)

Factoring Factoring is the process of expressing an algebraic expression as a product of other algebraic expressions. This is accomplished by factoring out common terms. Example: In the expression, all three terms have a common term 2a that can be factored out. Hence, we can rewrite this expression as or.

Exercise 5 Factor the following expressions: a) b)

Factoring Quadratics Of particular interest are determining the factors of second-degree polynomials with integer coefficients of the form. The factors of such expressions are always of the form where ac = p, bd = r, and ad + bc = q. Since only a limited number of choices are possible based on these relationships, the best way to determine these factors is often through trial and error. We will see this in the next two exercises.

Exercise 6 Factor the following expressions: a) b)

Roots of Quadratic Equations The roots of a quadratic equation are the values of x that satisfy that equation. Exercise 7 Find the roots of the following equations: a) b)

Roots of Quadratic Equations When it is difficult to find the roots of a quadratic, we can use the quadratic equation. The solutions of the equation are given by. Exercise 8 Find the roots of the following equation. Notice that there are no two integer numbers which multiply to –8 and add to 3.

Solving Systems of Equations Given n equations and n unknowns, suppose we wish to find the value of these unknowns (assume a solution exists). This can be done using one of two methods: elimination or substitution. Elimination involves adding/subtracting multiples of one equation from another in order to obtain a new equation that involves one less unknown. Substitution involves expressing one variable as a combination of the remaining variables for substitution into the remaining equations.

Exercise 9 Solve the following system of equations using (i) elimination and (ii) substitution:

Exercise 10 Solve the following system of equations using any method you wish.