intermediate algebra final exam review...

Int. Alg. Final Exam Review Sheet December 2007 of 24

Section R.3: Operations on Signed Numbers

To find the least common denominator:1. Factor each denominator.2. Write down the common factor(s), and then copy any additional factors.3. Multiply those numbers; the product is the least common denominator.4. Convert each rational number to an equivalent fraction with the LCD as denominator.

The Distributive Property of Real Numbers:If a, b, and c are real numbers, then

Section R.4: Order of OperationsExponentsRepeated multiplication can be written more efficiently using the notation of exponents:

Many calculators and computer languages use the “caret” symbol to denote exponents:

Order of Operations:1. Evaluate expressions within parentheses first. Begin with the innermost parentheses and work outward.2. Evaluate expressions involving exponents next, working from left to right.3. Evaluate expressions involving multiplication and division next, working from left to right.4. Evaluate expressions involving addition and subtraction next, working from left to right.

Section R.5: Algebraic ExpressionsSimplifying Algebraic Expressions: by Combining Like Terms:

by removing parentheses using the distributive property:


Section 1.1: Linear EquationsWe write an equation when we know what we want the answer to a multi-step to calculation to turn out to be.

1. Key fact that allows us to solve equations: Properties of Equality (i.e., the equality remains true if you do the same thing to both sides of the equation.): An equation can be transformed into an equivalent equation by adding or subtracting the same quantity on both sides of the equation, or by multiplying or dividing both sides of the equation by the same nonzero quantity.

2. The goal in manipulating equations when we solve them is to isolate the variable on one side of the equation.

3. When solving equations, proceed opposite the order of operations:a. Simplify both sides of the equation as much as possible by combining like terms and removing

parentheses suing the distributive property.b. Look for something to add or subtract to both sides.c. Look for something to multiply or divide both sides.d. Look for an exponent to raise both sides to.

Section 1.2:An Introduction to Problem Solving

English Mathis, was, are, yields, equals, gives, results in, is equal to, is equivalent to

=

product of, of (with a fraction or %)

sum of +difference -x more than y x + yx less than y y - xincrease +decrease -twice x or 2 times x 2xn times x nxTwo consecutive integers n, n+1

Area of a rectangle: A = Width LengthPerimeter = sum of the lengths of the sides of a shapeRate Time = AmountDistance = rate timeMixture Problems: Portion from Item A + Portion from Item B = Total

Percentage problems:

Problem Solving with Mathematical Models1. Identify what you are looking for.


2. Give names to the unknowns.3. Translate the problem into the language of mathematics. Use pictures to help you when possible.

Solve the equation.4. Check the reasonableness of your answer.5. Answer the original question.

Section 1.3: Using Formulas to Solve Problems

Some Geometric FormulasShape FormulaSquare Area

Perimeter Rectangle Area

Perimeter Triangle

Area

Perimeter Trapezoid

Area

Perimeter Parallelogram Area

Perimeter Circle Area

Circumference Cube Volume

Surface Area Rectangular Solid Volume

Surface Area Sphere

Volume

Surface Area Right Circular Cylinder Volume

Surface Area Cone

Volume


Section 1.4: Linear InequalitiesProperties of Inequalities: An inequality can be transformed into an equivalent inequality by adding or subtracting any quantity to both sides, or multiplying or dividing by any positive quantity. If both sides are multiplied or divided by a negative quantity, then the inequality symbol gets reversed.

Section 1.5: Compound InequalitiesIntersection of two sets: all the elements in both sets.Union of two sets: only the elements common to both sets.Compound Inequalities Using and: the solution is the intersection of the two sets.Compound Inequalities Using or: the solution is the union of the two sets.

Section 1.6: Absolute Value Equations and InequalitiesTo solve absolute value equations, use the facts that: is equivalent to u = a or u = -a.

is equivalent to u = v or u = -v.

To solve absolute value inequalities, use the facts that: is equivalent to is equivalent to is equivalent to or is equivalent to or

Section 2.1: Rectangular Coordinates and Graphs of EquationsTo graph a linear equation by plotting points, make a table with two rows of values, plot the points, then draw the line.

Graph Table:x y0 -52 -1

Plot points:

x

yDraw line:

x

y


To graph a nonlinear equation by plotting points, make a table with many rows of values, plot the points, then draw a smooth curve connecting the points.Graph y = 0.5x2 + 1.

x y-3 5.5-2 3.0-1 1.50 1.01 1.52 3.03 5.5

x

y

An x-intercept of a graph is the x-coordinate of a point on the graph that crosses or touches the x-axis.A y-intercept of a graph is the y-coordinate of a point on the graph that crosses or touches the y-axis.

x

y

y = x^4 - 5x^2 + 4

This graph has x-intercepts at x = -2, x = -1, x = +1, and x = +2, and a y-intercept at y = 4.

Section 2.2: RelationsA relation is a “link” from elements of one set to elements of another set.If x and y are two elements in these set, and if a relation exists between x and y, then we say:x corresponds to y, or y depends on x, and we write x y. We may also write a relation where y depends on x as an ordered pair (x, y).The domain of a relation is the set of all inputs to the relation.The range of a relation is the set of all outputs of the relation.

Section 2.3: An Introduction to Functions

Section 2.4: Functions and Their Graphs


Section 3.1: Linear Equations and Linear Functions

Definition: Linear EquationA linear equation (in two variables) is an equation of the form

Ax + By = CWhere A, B, and C are real numbers and A and B both cannot be zero. This is called the standard form for the equation of a line.

To graph a linear equation using intercepts:1. Let y = 0 and solve for x (for the x-intercept)2. Let x = 0 and solve for y (for the y-intercept)

Equation of a Vertical LineA vertical line is given by an equation of the form

x = aWhere a is the x-intercept.

Equation of a Horizontal LineA horizontal line is given by an equation of the form

y = bWhere b is the y-intercept.

Section 3.2: Slope and Equations of LinesDefinition: SlopeThe slope m between two points with coordinates (x1, y1) and (x2, y2) is defined by the formula

.

If x1 = x2, then the line between the two points is a vertical line, and the slope m is undefined.

Properties of Slope:m > 0 the line slants upward from left to rightm < 0 the line slants downward from left to rightm = 0 the line slants is horizontalm = undefined the line is vertical

Point-Slope Form of a LineIf you know the slope of a line (m) and one point on that line (x1, y1), then the equation of the line is given by the equation

(Point-Slope Form of a Line)An equation of a line L with slope m and y-intercept b has a slope-intercept form of

.


Equation of a line from two points:1) Find the slope.2) Use the point-slope form.

Section 3.3: Parallel and Perpendicular LinesDefinition: Relationship between the slopes of parallel linesTwo nonvertical lines are parallel if and only if their slopes are equal and they have different y-intercepts. Vertical lines are parallel if they have different x-intercepts.

Definition: Relationship between the slopes of perpendicular linesTwo nonvertical lines are perpendicular if and only if the product of their slopes is -1. Alternatively, their slopes are negative reciprocals of one another. Vertical lines are perpendicular to horizontal lines.

Section 3.4: Linear Inequalities in Two Variables Step 1: Write the inequality as an equality, then graph the equation using a dashed line if it is a strict inequality (< or >), or using a solid line if it is not a strict inequality ( or ). Step 2: Pick a test point and see if the ordered pair satisfies the inequality. If it does satisfy, shade the half of the plane on the side of the line containing the point. If it does not satisfy, shade the other half.

Section 3.5: Building Linear ModelsYou should be able to write equations for linear models given a verbal description, a direct variation, or a set of data.

Section 4.1: Systems of Linear Equations in Two VariablesSolution by substitution:

1. Solve one equation for one of the variables.2. Substitute that expression for that variable into the other equation.3. Solve for the remaining variable.4. Back-substitute into the first equation to get the value for the other variable.

Section 4.2: Problem Solving: Systems of Two Linear Equations Containing Two UnknownsYou should be able to write out a model for a real-world situation involving two equations in two unknowns, then solve that system to get an answer.


Section 4.3: Systems of Linear Equations in Three VariablesExample of a system of three equations in three unknowns that is in triangular form:

Notice: the name triangular form comes from the “blank” triangular space in the lower left corner due to no x or y variables. Also, this system is really easy to solve using back-substitution:

Steps for Solving a System of Three Linear Equations in Three Unknowns Using Elimination1. Eliminate the same one variable from two of the equations using elimination.2. Use elimination to remove a second variable from those two equations.3. Solve for the remaining variable.4. Substitute the answer for that variable into the remaining equations.

Section 4.4: Using Matrices to Solve SystemsExample of a system of three equations in three unknowns that is in triangular form:

Use row operations to convert the matrix into triangular form.


Section 4.5: Determinants and Cramer's RuleDefinition: Determinant of a 2 2 matrix

Suppose that a, b, c, and d are real numbers. The determinant of the 2 2 matrix , written as ,

is .

Cramer’s Rule for solving a system of two linear equations in two unknowns

The solution to the system of equations is given by

and

Provided that

Definition: Determinant of a 3 3 matrix

The determinant of the 3 3 matrix , written as , is calculated using the

determinants of 2 2 matrices as follows:

.

Cramer’s Rule for solving a system of three linear equations in three unknowns

The solution to the system of equations

with

, , , and

is given by

, , and .


Section 4.6: Systems of Linear Inequalities

Graph the system .

Section 5.1: Adding and Subtracting PolynomialsTo add polynomials, combine like terms.To subtract polynomials, combine like terms.A polynomial function is a function whose rule is a polynomial. The domain of a polynomial function is all real numbers. The degree of a polynomial function is the value of the largest exponent on the variable.

Section 5.2: Multiplying PolynomialsExtended form of the Distributive Property:

To multiply polynomials, use distribution, or multiply each term in the first polynomial by each term in the second polynomial.

Example: (x2 + 3x + 9)(x2 – 2x +3) = x4 –2x3 +3x2

+3x3 -6x2 +9x+9x2 -18x +27

=x4 +x3 +6x2 -9x +27

Section 5.3: Dividing Polynomials; Synthetic DivisionDividing a polynomial by a monomial: Divide the monomial into each term of the polynomial, and cancel when possible.

Dividing a polynomial by a polynomial using long division: Long division of polynomials is a lot like long division of numbers:

a. Arrange divisor and dividend around the dividing symbol, and be sure to write them in descending order of powers with all terms explicitly stated.

b. Divide leading terms, then multiply and subtract.c. Repeat until a remainder of order less than the divisor is obtained.

Example:


Dividing a polynomial by a polynomial using synthetic division: THIS IS A SHORTCUT THAT ONLY WORKS WHEN THE DEGREE OF THE DIVISOR IS 1 (I.E., THE DIVISOR IS x – c) !!!Synthetic division is a shorthand way to divide a polynomial by a linear factor.:

a. Write c outside the bar and the coefficients of the dividend inside the bar.b. Bring the leading coefficient straight down.c. Compute c times the number in the bottom row, and write the answer in the middle row to the

right.d. Add and repeat until all coefficients are used up.

The Remainder TheoremIf the polynomial P(x) is divided by x – c, then the remainder is the value P(c).

The Factor TheoremIf P is a polynomial function, then x – c is a factor of P if and only if P(c) = 0.(This can be used to see if a divisor divides evenly into a dividend quickly)

Section 5.4: Greatest Common Factor; Factoring by GroupingFactoring the greatest common factor:

1. Identify the greatest common factor (GCF) in each term.2. Rewrite each term as the product of the GCF and the remaining factor.3. Use the Distributive Property to factor out the GCF.4. Use the Distributive Property to verify that the factorization is correct.

Example:


Factoring by grouping:1. Group the terms with common factor. You may need to rearrange the terms.2. In each grouping, factor out the common factor.3. Factor out the common factor that remains.4. Check your work. .

Example:

Section 5.5: Factoring Trinomials1. Factoring x2 + qx + p:

a. Look for factors of p that sum to qb. Example: x2 – 4x – 12 =

i. factors of 12 are: 1, 2, 3, 4, 6, 12ii. the factors that add up to –4 are: -4 = –6 and +2

iii. x2 – 4x – 12 = (x – 6)(x + 2)2. Factoring Ax2 + Bx + C:

a. List factors of AC that add to the coefficient B, then write the middle term as a sum of those factors

b. Factor by groupingc. Example: 2x2 – 9x – 18 =

i. (2)(-18) = -36ii. factors of -36 that add to -9 are: -9 = -12 and + 3

iii. 2x2 – 9x – 18 = 2x2 – 12x + 3x – 18 = 2x(x – 6) + 3(x – 6) = (2x + 3)(x – 6)

Section 5.6: Factoring Special Products1. Difference of Two Squares: a2 – b2 = (a + b)(a – b)

a. Example: 9x2 – 81 = (3x + 3)(3x – 3)2. Square of Sum: a2 + 2ab + b2 = (a + b)2

a. Example: x2 + 12x + 36 = (x + 6)2

3. Square of Difference: a2 – 2ab + b2 = (a – b)2

a. Example: x2 - 2x + 1 = (x – 1)2

4. Difference of Cubes: a3 – b3 = (a – b)(a2 + ab + b2)a. Example: x3 – 8 = (x – 2)(x2 + 2x + 4)

5. Sum of Cubes: a3 + b3 = (a + b)(a2 – ab + b2)a. Example: 27x3 + 64 = (3x + 4)(9x2 – 12x + 16)

Section 5.7: Factoring: A General StrategySee the book…


Section 5.8: Polynomial EquationsThe Zero-Product PropertyIf the product of two numbers is zero, then at least one of the numbers is zero. That is, if ab = 0, then a = 0 or b = 0, or both a and b are 0.Steps for solving any polynomial equation (quadratic or higher):

1. Expand the polynomial equation (if needed), and collect all terms on one side; combine like terms.2. Factor the polynomial on the one side.3. Set each factor from step 2 equal to zero (this is justified by the zero-product property rule).4. Solve each first degree equation for the variable.5. Check answers in original equation.

Example: x3 – 4x2 = 12xx3 – 4x2 – 12x = 0x(x3 – 4x2 – 12)= 0x(x – 6)(x + 2) = 0

x = 0 or x – 6 = 0 or x + 2 = 0x = 0 x = 6 x = -2

Section 6.1: Multiplying and Dividing Rational ExpressionsA rational expression is simplified when all common factors in the numerator and denominator have been cancelled.Example:

To multiply rational expressions, use the arithmetic rule

Example:

To divide rational expressions, use the arithmetic rule (i.e., invert and multiply)

Example:


Section 6.2: Adding and Subtracting Rational ExpressionsWhen rational expressions have common denominators, they can be added or subtracted by simply adding or subtracting the numerators (like with arithmetic fractions).To find the least common denominator of a rational expression, factor each denominator, then list all the common factors and all the uncommon factors.To add or subtract rational expressions, find the LCD, then write each rational expression with the common denominator, then add or subtract numerators.Example:

Section 6.3: Complex Rational ExpressionsKeys:

1. Method #1: Simplify numerator and denominator to a single rational expression, then invert and multiply.

2. Method #2: Multiply numerator and denominator by the L.C.D. of all fractions…

Section 6.4: Rational EquationsSteps for Solving a Rational Equation

1. Determine the domain of the rational equation.2. Determine the LCD of all the denominators.3. Multiply both sides of the equation by the LCD and simplify each side of the equation into polynomials.4. Solve the resulting polynomial equation (using factoring and the zero-product property if needed).5. Verify your solutions using the original equation. Make sure they are in the domain of the equation (i.e.,

that they are not extraneous solutions).


Section 6.5: Rational InequalitiesSteps for Solving a Rational Inequality1. Write the inequality so it is a single rational expression on one side of the inequality and zero on the other

side. Completely factor the numerator and denominator of the rational expression.2. Determine all numbers that make the factors of the rational expression zero.3. Use those zeros to separate the real number line into intervals.4. Choose a test point in each interval, and determine the sign of the rational expression at that test point. If

the sign for a test value matches the inequality, then the entire interval containing the test point is a solution.

Section 6.6: Models Involving Rational Expressions


Section 7.1: n th Roots and Rational Exponents Definition: The principal nth root of a number

, where n is an integer greater than or equal to 2, computes to a number b such that if , then .

If n is an even number, then a and b must be positive.If n is an odd number, then a and b can be any real number.

n 2^n 3^n 4^n 5^n 6^n 7^n 8^n 9^n 10^n2 4 9 16 25 36 49 64 81 1003 8 27 64 125 216 343 512 729 1,0004 16 81 256 625 1,296 2,401 4,096 6,561 10,0005 32 243 1,024 3,125 7,776 16,807 32,768 59,049 100,0006 64 729 4,096 15,625 46,656 117,649 262,144 531,441 1,000,0007 128 2,187 16,384 78,125 279,936 823,543 2,097,152 4,782,969 8 256 6,561 65,536 390,625 1,679,616 5,764,801 9 512 19,683 262,144 1,953,125 10 1,024 59,049 1,048,576 9,765,625

n (-2)^n (-3)^n (-4)^n (-5)^n (-6)^n (-7)^n (-8)^n (-9)^n (-10)^n2 4 9 16 25 36 49 64 81 1003 -8 -27 -64 -125 -216 -343 -512 -729 -1,0004 16 81 256 625 1,296 2,401 4,096 6,561 10,0005 -32 -243 -1,024 -3,125 -7,776 -16,807 -32,768 -59,049 -100,0006 64 729 4,096 15,625 46,656 117,649 262,144 531,441 1,000,0007 -128 -2,187 -16,384 -78,125 -279,936 -823,543 -2,097,152 -4,782,969 8 256 6,561 65,536 390,625 1,679,616 5,764,801 9 -512 -19,683 -262,144 -1,953,125 10 1,024 59,049 1,048,576 9,765,625

Section 7.2: Simplifying Expressions Using the Laws of Exponents

Formulas for integer exponents:


Section 7.3: Simplifying Radical ExpressionsBig Idea: A radical is in simplest form when:

As many powers as possible are pulled out of the radical.The order of the radical is as low as possible.There are no radicals in the denominator.

Steps to simplify a radical expression:Write each factor of the radicand as the product of two factors, one of which is a perfect power of the

radicand.Write the radicand as the product of two radicals (using the product property of radicals).Take the nth root of the perfect power.

Section 7.4: Adding, Subtracting, and Multiplying Radical ExpressionsBig Idea: Radicals can only be added or subtracted when they are similar. Similar radicals have the same order and same radicand.

If the radicals are not similar, then they can not be combined by addition or subtraction.Example of similar radicals that can be added:

Example of radicals that are not similar because of different orders and thus can not be added: Example of radicals that are not similar because of different radicands and thus can not be added:

Radicals that do not look similar may in fact be similar if they are simplified first:


Sections 7.5: Rationalizing Radical ExpressionsBig Idea: A radical is in simplest form when:

As many powers as possible are pulled out of the radical.The order of the radical is as low as possible.There are no radicals in the denominator. (This is simply because before calculators, it was a pain to

divide by a radical long-hand).

To eliminate radical factors in the denominator, multiply numerator and denominator by an appropriate power of the denominator to make the denominator a perfect power of the order of the radical. This is called rationalizing the denominator.To eliminate a sum of square roots in the denominator, multiplying numerator and denominator by the conjugate of the denominator. This trick is based on the difference of squares: (a + b)(a – b) = a2 – b2.

Section 7.6: Functions Involving RadicalsDomain of a radical function:If the index of the radical is even, then the domain is all numbers that make the radicand non-negative.If the index of the radical is odd, then the domain is all real numbers.

Section 7.8: The Complex Number SystemDefinition: The Imaginary UnitThe imaginary unit, denoted by the symbol i, is the number whose square root is –1. That is, i2 = –1.Taking the square root of both sides, we can see that

Evaluating Square Roots of negative NumbersIf N is a positive real number, then the principal square root of –N is calculated as:

Definition: Complex NumbersComplex numbers are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit. The number a is called the real part of the complex number, and the number b is called the imaginary part of the complex number.

Sum of Complex NumbersTo add complex numbers, add reals to reals, and add imaginaries to imaginaries:

(a + bi) + (c + di) = (a + c) + (b + d)iDifference of Complex NumbersTo subtract complex numbers, subtract reals from reals, and subtract imaginaries from imaginaries:

(a + bi) – (c + di) = (a – c) + (b – d)iNote: Adding or subtracting complex numbers is just like combining like terms.

Product of Complex NumbersTo multiply complex numbers, use the Distributive Property (or the FOIL method), then collect like terms:

(a + bi)(c + di) = (ac – bd) + (ad + bc)i


Definition: Complex Conjugate: For a complex number a + bi, its conjugate is defined as a – bi.

Quotient of Complex NumbersTo divide complex numbers, multiply top and bottom by the complex conjugate of the denominator. Note that the denominator can be simplified using the difference of squares formula.

Sections 8.1: Solving Quadratic Equations by Completing the SquareThe Square Root Property:If x2 = p, then or .Completing the SquareTo obtain a perfect square trinomial:

Identify the coefficient of the first-degree term. Multiply this coefficient by , then square the result. That is,

if you have ax2 + bx + c, a value of will complete the square.

Solving a Quadratic Equation by Completing the Square:Write the quadratic equation as x2 + bx = -c.Complete the square on x2 + bx by making it a perfect square trinomial. Don’t forget to add the same

amount to the right hand side.Factor the perfect square trinomial.Solve using the Square Root Property.

Section 8.2: Solving Quadratic Equations by the Quadratic FormulaThe Quadratic Formula:

If ax2 + bx + c = 0, then

The DiscriminantThe discriminant for a quadratic equation is the quantity . Its value tells us the number and type of solution to expect from the quadratic formula.

Discriminant Number of Solutions

Type of Solutions Practice Example

Positive and a perfect square

2 Rational 1

Positive and not a perfect square

2 Irrational 2

Zero 1 (a repeated root) Rational 3Negative 2 Complex 4


Section 8.3: Solving Equations Quadratic in FormTo solve x4 – 4x2 – 12 = 0, let u = x2. Then the equation becomes u2 – 4u – 12 = 0, so

which means that

Section 8.4: Graphing Quadratic Equations Using TransformationsAdding or subtracting a constant from every occurrence of the variable x shifts the graph left or right.The graph of y = f(x – h) simply shifts the graph of y = f(x) by h units to the right.

The graph of y = f(x + h) simply shifts the graph of y = f(x) by h units to the left.

Multiplying a function by a constant makes the graph steeper or even reflects it about the x-axis.


Adding or subtracting a constant from the entire function shifts the graph up or down.The graph of y = f(x) + k simply shifts the graph of y = f(x) by k units up.

The graph of y = f(x) – k simply shifts the graph of y = f(x) by k units down.

To quickly sketch the graph of a quadratic equation of the form y = a(x – h)2 + k:Locate the point (h, k). This is the vertex.If a is positive, the parabola opens up; if a is negative, the parabola opens down.If |a| > 1, the parabola is “steep”. If |a| < 1, the parabola is “squashed.”Draw the parabola with the correct open up/down and steepness or squashedness.

Section 8.5: Graphing Quadratic Equations Using Properties

To quickly and ACCURATELY sketch the graph of a quadratic equation of the form y = a(x – h)2 + k:

Locate the point . This is the vertex.

Draw a dashed line at . This is the symmetry axis of the parabola.

Calculate and plot the zeros of the function (if there are any).Compute a couple points on either side of the symmetry axis, then sketch the curve.


Section 9.2: Exponential FunctionsThe exponential function is a constant number raised to an exponent that is a variable:

y = bx

Facts about the exponential function:The constant number b is called the base.The base must be greater than zero (b > 0).Some common bases are 2, 10, and e.The base can’t be equal to one (b 1).The variable x, which is the exponent, can be any real number (i.e., the domain of the function is all real

numbers).The dependent variable y will take on all values greater than zero (i.e., the range is y > 0).The graph always passes through the point (0, 1), because any number raised to the power of zero equals

one.The graph also always contains the points (1, b) and (-1, 1/b).The graph of the function increases if b > 1, while the graph decreases if 0 < b < 1, as shown below:

This is for b > 1 This is for 0 < b < 1

Definition of e:

The number e is defined as the number that the expression approaches as n gets bigger without bound.

Solving exponential equations:If au = av, then u = v.

Compound Interest Formula:The amount A in a bank account for a principal P compounded at an annual interest rate r n times per year for t

years is:


Section 9.3: Logarithmic FunctionsBig Idea: The logarithmic function “undoes” the exponential function.

x = by is equivalent to y = logb x.

The first formula is in “exponential form,” while the second equation is in “logarithmic form.”The logarithm is just an exponent. In fact, it helps sometimes to not to say the word logarithm, but instead “the exponent on b that gives x for an answer.”

Facts about the logarithmic function for b > 1: The constant number b is called the base.Some common bases are 2, 10, and e.

a. When the base is 10, we write using shorthand: log10 100 = log 100b. When the base is e, we write using shorthand: loge 7 = ln 7

The answer generated by the logarithmic function can be any real number (i.e., the range of the function is all real numbers - < y < ).

The number you plug into the logarithmic function can only be positive (i.e., the domain is x > 0).The graph always passes through the point (1, 0), because any number raised to the power of zero equals

one.There are no There are no y-intercepts.The logarithm of a fraction between zero and one is negative.The logarithm of a number greater than one is positive.The graph of the function increases, as shown below:

Solving logarithmic equations:Convert the equation to exponential form, and go from there.


Definition: The loudness L, measured in decibels, of a sound intensity x measured in units of watts per square meter is:

Section 9.4: Properties of Logarithms

Exponent Rule Corresponding Logarithm Rule

Change of Base Formula:

Determining the Domain of a variable:Assume the domain is all real numbers.Rule out any numbers that would make the denominator equal to zero.Rule out any numbers that would make the radicand negative for even radicals.Rule out any numbers that would make the argument of a logarithm negative.For word problems, rule out any numbers that do not make sense relative to the word problem (like negative

costs or distances or weights, etc.)

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