Solving Exponential Equations

Size: px
Start display at page:

Transcription

1 Solving Exponential Equations Deciding How to Solve Exponential Equations When asked to solve an exponential equation such as x + 6 = or x = 18, the first thing we need to do is to decide which way is the best way to solve the problem. Some exponential equations can be solved by rewriting each side of the equation using the same base. Other exponential equations can only be solved by using logarithms. How do we decide what is the best way to solve an exponential equation? The key is to look at the base of the exponential equation and determine if the each side of the problem can be rewritten using the same base. If we consider the problem x + 6 =, the base of the exponent is and we need to decide if we can rewrite the number using only the number. In this case it is possible to write the number using only s, = =. This means that the best way to solve the problem x + 6 = is to rewrite the problem using the base. If we consider the problem x = 18, the base of the exponent is and we need to decide if we can rewrite the number 18 using only the number. In this case it is impossible to write the number 18 using only s, 18 = =. This means that the best and only way to solve the problem x = 18 is to Solving Exponential Equations with the Same Base After deciding the best way to solve an exponential equation is by rewriting each side of the equation using the same base, what is next? Solving Exponential Equations with the Same Base If B M = B N, then M = N. This statement simply says that if the bases are the same then the exponents must be the same. To solve an exponential equation with the same base, first we need to rewrite the problem using the same base and after getting the bases the same we can drop the bases and set the Let s finish solving the problem x + 6 = from before. In this problem we have already seen that both and can be written using the number. If we continue from there we get: x+ 6 = x + 6 = x = 1 Rewrite the problem using the same base. Finish solving by subtracting 6 from each side. Therefore, the solution to the problem x + 6 = is x = 1.

2 There is another example, solve 9 x = 7. 9 x = 7 Determine if 9 and 7 can be written using the same base. In this case both 9 and 7 can be written using the base. ( ) x = 4x = 4x = x = Rewrite the problem using the same base. Use the properties of exponents to simplify the exponents, when a power is raised to a power, we multiply the powers. Finish solving by adding to each side and then dividing each side by 4. Therefore, the solution to the problem 9 x = 7 is x =. Now that we have looked at a couple of examples of solving exponential equations with the same base, let s list the steps for solving exponential equations that have the same base. Steps for Solving Exponential Equations with the Same Base Step 1: Determine if the numbers can be written using the same base. If so, go to Step. If not, stop and use Steps for Solving an Exponential Equation with Different Bases. Step : Rewrite the problem using the same base. Step : Use the properties of exponents to simplify the problem. Step 4: Once the bases are the same, drop the bases and set the Step : Finish solving the problem by isolating the variable. Solving Exponential Equations with the Different Bases After deciding the only way to solve an exponential equation is to use logarithms, what is next? The next step is to take the common logarithm or natural logarithm of each side. By taking the logarithm of each side, we can use the properties of logarithms, specifically property from our list of properties, to rewrite the exponential problem as a multiplication problem. After changing the problem from an exponential problem to a multiplication problem using the properties of logarithms we will be able to finish solving the problem.

3 Let s finish solving the problem x = 18 from before. In this problem we have already seen that it is impossible to rewrite the numbers and 18 using the same base, so we must Continuing on here is what we get: x log( ) = log(18) (x )(log ) = log18 log18 x = log x x Divide each side by log. Use a calculator to find log 18 divided by log. Round the answer as appropriate, these answers will use 6 decimal places. Finish solving the problem by adding to each side and then dividing each side by. Therefore, the solution to the problem x = 18 is x The directions say, You will get the same answer no matter which logarithm you use, it is a matter of personal preference. The only time that you should specifically use a natural logarithm is when dealing with the base e. Here is another example, solve 8 4x + 1 = 0. 4x+ 1 8 = 0 4x+ 1 log(8 ) = log(0) (4x + 1)(log 8) = log 0 log 0 4x + 1 = log8 4x x Determine if 8 and 0 can be written using the same base. In this case 8 and 0 cannot be written using the same base, so we must Divide each side by log 8. Use a calculator to find log 0 divided by log 8. Round the answer as appropriate, these answers will use 6 decimal places. Finish solving the problem by subtracting 1 from each side and then dividing each side by 4. Therefore, the solution to the problem 8 4x + 1 = 0 is x Now that we have looked at a couple of examples of solving exponential equations with different bases, let s list the steps for solving exponential equations that have different bases.

4 Solving Exponential Equations with Different Bases Step 1: Determine if the numbers can be written using the same base. If so, stop and use Steps for Solving an Exponential Equation with the Same Base. If not, go to Step. Step : Step : Use the properties of logarithms to rewrite the problem. Specifically, use Property which y says log x = ylog x. a Step 4: Divide each side by the logarithm. a Step : Use a calculator to find the decimal approximation of the logarithms. Step 6: Finish solving the problem by isolating the variable. Examples Now let s use the steps shown above to work through some examples. These examples will be a mixture of exponential equations with the same base and exponential equations with different bases. Example 1: Solve x + 7 = 11 x+ 7 = 11 x+ 7 log( ) = log(11) Determine if and 11 can be written using the same base. In this case and 11 cannot be written using the same base, so we must (x + 7)(log ) = log 11 log11 x + 7 = log x x Divide each side by log. Use a calculator to find log 11 divided by log. Round the answer as appropriate, these answers will use 6 decimal places. Finish solving the problem by subtracting 7 from each side and then dividing each side by. Therefore, the solution to the problem x + 7 = 11 is x

5 Example : Solve x 1 = 1 x + 4 x 1 x+ 4 = 1 ( ) = ( ) x 1 x+ 4 Determine if and 1 can be written using the same base. In this case both and 1 can be written using the base. Rewrite the problem using the same base. 4x 9x+ 1 = 4x = 9x + 1 Use the properties of exponents to simplify the exponents, when a power is raised to a power, we multiply the powers. 14 x = 14 Therefore, the solution to the problem x 1 = 1 x + 4 is x =. Example : Solve e 4x 9 = 6 Finish solving by adding to each side, subtracting 9x from each side, and then dividing each side by. 4x 9 e = 6 4x 9 ln(e ) = ln(6) (4x 9)(ln e) = ln 6 (4x 9)(1) = ln 6 4x x.68 Determine if e and 6 can be written using the same base. In this case e and 6 cannot be written using the same base, so we must In this case, we should use the natural logarithm because the base is e. Use the properties of logarithms to find ln e. Property states that ln e = 1. Use a calculator to find ln 6. Round the answer as appropriate, these answers will use 6 decimal places. Finish solving the problem by adding 9 to each side and then dividing each side by 4. Therefore, the solution to the problem e 4x 9 = 6 is x.68.

6 Example 4: Solve 7 x + = 1 x+ 7 = 1 x+ log(7 ) = log(1) Determine if 7 and 1 can be written using the same base. In this case 7 and 1 cannot be written using the same base, so we must (x + )(log 7) = log 1 log1 x + = log 7 x x Divide each side by log 7. Use a calculator to find log 1 divided by log 7. Round the answer as appropriate, these answers will use 6 decimal places. Finish solving the problem by subtracting from each side and then dividing each side by. Therefore, the solution to the problem 7 x + = 1 is x Example : Solve 8 x 1 16 x 8 x 1 16 x 4 ( ) = ( ) x x Determine if 8 and 1/16 can be written using the same base. In this case both 8 and 1/16 can be written using the base. Rewrite the problem using the same base. Note that 1/16 = 1/ 4 = 4. 6x 9 4x+ 8 = 6x 9 = 4x x = 10 Use the properties of exponents to simplify the exponents, when a power is raised to a power, we multiply the powers. Finish solving by adding 9 to each side, adding 4x to each side, and then dividing each side by 10. Therefore, the solution to the problem 8 x 1 16 x is 17 x =. 10

7 Addition Examples If you would like to see more examples of solving exponential equations, just click on the link below. Additional Examples Practice Problems Now it is your turn to try a few practice problems on your own. Work on each of the problems below and then click on the link at the end to check your answers. Problem 1: Solve: Problem : Solve: Problem : Solve: Problem 4: Solve: Problem : Solve: Problem 6: Solve: x+ 4x 4 = 8 8 x e = 68 6x+ 7 = x x = 87 x 4e = 19 x+ 1 Solutions to Practice Problems

Solving Compound Interest Problems

Solving Compound Interest Problems What is Compound Interest? If you walk into a bank and open up a savings account you will earn interest on the money you deposit in the bank. If the interest is calculated

SOLVING EQUATIONS WITH EXCEL

SOLVING EQUATIONS WITH EXCEL Excel and Lotus software are equipped with functions that allow the user to identify the root of an equation. By root, we mean the values of x such that a given equation cancels

8.7 Exponential Growth and Decay

Section 8.7 Exponential Growth and Decay 847 8.7 Exponential Growth and Decay Exponential Growth Models Recalling the investigations in Section 8.3, we started by developing a formula for discrete compound

Pre-AP Algebra 2 Unit 9 - Lesson 2 Introduction to Logarithms Objectives: Students will be able to convert between exponential and logarithmic forms of an expression, including the use of the common log.

Rational Exponents. Squaring both sides of the equation yields. and to be consistent, we must have

8.6 Rational Exponents 8.6 OBJECTIVES 1. Define rational exponents 2. Simplify expressions containing rational exponents 3. Use a calculator to estimate the value of an expression containing rational exponents

Section 4.5 Exponential and Logarithmic Equations

Section 4.5 Exponential and Logarithmic Equations Exponential Equations An exponential equation is one in which the variable occurs in the exponent. EXAMPLE: Solve the equation x = 7. Solution 1: We have

3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions.

Functions - Exponential Functions

0.4 Functions - Exponential Functions Objective: Solve exponential equations by finding a common base. As our study of algebra gets more advanced we begin to study more involved functions. One pair of

Section 1. Logarithms

Worksheet 2.7 Logarithms and Exponentials Section 1 Logarithms The mathematics of logarithms and exponentials occurs naturally in many branches of science. It is very important in solving problems related

How To Understand Algebraic Equations

Please use the resources below to review mathematical concepts found in chemistry. 1. Many Online videos by MiraCosta Professor Julie Harland: www.yourmathgal.com 2. Text references in red/burgundy and

eday Lessons HSCC Precalculus Logarithims F-LE 4, BF-B 5 11/2014 E-Lesson 1

eday Lessons HSCC Precalculus Logarithims F-LE 4, BF-B 5 11/2014 E-Lesson 1 Enclosed are the E-Day assignments required to make up the 3 calamity days missed during the 2014-2015 school year for High School

Pre-Session Review. Part 2: Mathematics of Finance

Pre-Session Review Part 2: Mathematics of Finance For this section you will need a calculator with logarithmic and exponential function keys (such as log, ln, and x y ) D. Exponential and Logarithmic Functions

Negative Integer Exponents

7.7 Negative Integer Exponents 7.7 OBJECTIVES. Define the zero exponent 2. Use the definition of a negative exponent to simplify an expression 3. Use the properties of exponents to simplify expressions

12.6 Logarithmic and Exponential Equations PREPARING FOR THIS SECTION Before getting started, review the following:

Section 1.6 Logarithmic and Exponential Equations 811 1.6 Logarithmic and Exponential Equations PREPARING FOR THIS SECTION Before getting started, review the following: Solve Quadratic Equations (Section

Logarithmic and Exponential Equations

11.5 Logarithmic and Exponential Equations 11.5 OBJECTIVES 1. Solve a logarithmic equation 2. Solve an exponential equation 3. Solve an application involving an exponential equation Much of the importance

Exponential & Logarithmic Equations

Exponential & Logarithmic Equations This chapter is about using the inverses of exponentials or logarithms to solve equations involving exponentials or logarithms. Solving exponential equations An exponential

MBA Jump Start Program

MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right

Solving Linear Equations in One Variable. Worked Examples

Solving Linear Equations in One Variable Worked Examples Solve the equation 30 x 1 22x Solve the equation 30 x 1 22x Our goal is to isolate the x on one side. We ll do that by adding (or subtracting) quantities

Equations. #1-10 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0

College Algebra Review Problems for Final Exam Equations #1-10 Solve for the variable 1. 2 1 4 = 0 6. 2 8 7 2. 2 5 3 7. = 3. 3 9 4 21 8. 3 6 9 18 4. 6 27 0 9. 1 + log 3 4 5. 10. 19 0 Inequalities 1. Solve

CHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises

CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =

Chapter 1: Order of Operations, Fractions & Percents

HOSP 1107 (Business Math) Learning Centre Chapter 1: Order of Operations, Fractions & Percents ORDER OF OPERATIONS When finding the value of an expression, the operations must be carried out in a certain

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.

MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 2-49x + 6 x - 6 A) 1, x 6 B) 8x - 1, x 6 x -

MULTIPLICATION AND DIVISION OF REAL NUMBERS In this section we will complete the study of the four basic operations with real numbers.

1.4 Multiplication and (1-25) 25 In this section Multiplication of Real Numbers Division by Zero helpful hint The product of two numbers with like signs is positive, but the product of three numbers with

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

General Exponent Rules: Exponents, Radicals, and Scientific Notation x m x n = x m+n Example 1: x 5 x = x 5+ = x 7 (x m ) n = x mn Example : (x 5 ) = x 5 = x 10 (x m y n ) p = x mp y np Example : (x) =

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

Chapter 7 - Roots, Radicals, and Complex Numbers

Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

LESSON EIII.E EXPONENTS AND LOGARITHMS

LESSON EIII.E EXPONENTS AND LOGARITHMS LESSON EIII.E EXPONENTS AND LOGARITHMS OVERVIEW Here s what ou ll learn in this lesson: Eponential Functions a. Graphing eponential functions b. Applications of eponential

2.6 Exponents and Order of Operations

2.6 Exponents and Order of Operations We begin this section with exponents applied to negative numbers. The idea of applying an exponent to a negative number is identical to that of a positive number (repeated

Algebra II New Summit School High School Diploma Program

Syllabus Course Description: Algebra II is a two semester course. Students completing this course will earn 1.0 unit upon completion. Required Materials: 1. Student Text Glencoe Algebra 2: Integration,

Factor Diamond Practice Problems

Factor Diamond Practice Problems 1. x 2 + 5x + 6 2. x 2 +7x + 12 3. x 2 + 9x + 8 4. x 2 + 9x +14 5. 2x 2 7x 4 6. 3x 2 x 4 7. 5x 2 + x -18 8. 2y 2 x 1 9. 6-13x + 6x 2 10. 15 + x -2x 2 Factor Diamond Practice

Example. L.N. Stout () Problems on annuities 1 / 14

Example A credit card charges an annual rate of 14% compounded monthly. This month s bill is \$6000. The minimum payment is \$5. Suppose I keep paying \$5 each month. How long will it take to pay off the

MPE Review Section III: Logarithmic & Exponential Functions

MPE Review Section III: Logarithmic & Eponential Functions FUNCTIONS AND GRAPHS To specify a function y f (, one must give a collection of numbers D, called the domain of the function, and a procedure

QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS Content 1. Parabolas... 1 1.1. Top of a parabola... 2 1.2. Orientation of a parabola... 2 1.3. Intercept of a parabola... 3 1.4. Roots (or zeros) of a parabola...

Session 29 Scientific Notation and Laws of Exponents. If you have ever taken a Chemistry class, you may have encountered the following numbers:

Session 9 Scientific Notation and Laws of Exponents If you have ever taken a Chemistry class, you may have encountered the following numbers: There are approximately 60,4,79,00,000,000,000,000 molecules

Higher Education Math Placement

Higher Education Math Placement Placement Assessment Problem Types 1. Whole Numbers, Fractions, and Decimals 1.1 Operations with Whole Numbers Addition with carry Subtraction with borrowing Multiplication

2.3 Solving Equations Containing Fractions and Decimals

2. Solving Equations Containing Fractions and Decimals Objectives In this section, you will learn to: To successfully complete this section, you need to understand: Solve equations containing fractions

Review of Scientific Notation and Significant Figures

II-1 Scientific Notation Review of Scientific Notation and Significant Figures Frequently numbers that occur in physics and other sciences are either very large or very small. For example, the speed of

PREPARATION FOR MATH TESTING at CityLab Academy

PREPARATION FOR MATH TESTING at CityLab Academy compiled by Gloria Vachino, M.S. Refresh your math skills with a MATH REVIEW and find out if you are ready for the math entrance test by taking a PRE-TEST

Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y)

Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = Last day, we saw that the function f(x) = ln x is one-to-one, with domain (, ) and range (, ). We can conclude that f(x) has an inverse function

PURSUITS IN MATHEMATICS often produce elementary functions as solutions that need to be

Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions 2007 Ron Doerfler http://www.myreckonings.com June 27, 2007 Abstract There are some of us who enjoy using our

Background Information on Exponentials and Logarithms

Background Information on Eponentials and Logarithms Since the treatment of the decay of radioactive nuclei is inetricably linked to the mathematics of eponentials and logarithms, it is important that

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

5.3 SOLVING TRIGONOMETRIC EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations

Math Placement Test Study Guide. 2. The test consists entirely of multiple choice questions, each with five choices.

Math Placement Test Study Guide General Characteristics of the Test 1. All items are to be completed by all students. The items are roughly ordered from elementary to advanced. The expectation is that

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

The gas can has a capacity of 4.17 gallons and weighs 3.4 pounds.

hundred million\$ ten------ million\$ million\$ 00,000,000 0,000,000,000,000 00,000 0,000,000 00 0 0 0 0 0 0 0 0 0 Session 26 Decimal Fractions Explain the meaning of the values stated in the following sentence.

Precalculus Orientation and FAQ

Precalculus Orientation and FAQ MATH 1011 (Precalculus) is a four hour 3 credit course that prepares a student for Calculus. Topics covered include linear, quadratic, polynomial, rational, exponential,

6. Differentiating the exponential and logarithm functions

1 6. Differentiating te exponential and logaritm functions We wis to find and use derivatives for functions of te form f(x) = a x, were a is a constant. By far te most convenient suc function for tis purpose

Lesson Plan -- Rational Number Operations

Lesson Plan -- Rational Number Operations Chapter Resources - Lesson 3-12 Rational Number Operations - Lesson 3-12 Rational Number Operations Answers - Lesson 3-13 Take Rational Numbers to Whole-Number

MATH BOOK OF PROBLEMS SERIES. New from Pearson Custom Publishing!

MATH BOOK OF PROBLEMS SERIES New from Pearson Custom Publishing! The Math Book of Problems Series is a database of math problems for the following courses: Pre-algebra Algebra Pre-calculus Calculus Statistics

Finance 197. Simple One-time Interest

Finance 197 Finance We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for

Welcome to Math 19500 Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 2013 An important feature of the following Beamer slide presentations is that you, the reader, move

Math Common Core Sampler Test

High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests

4.6 Exponential and Logarithmic Equations (Part I)

4.6 Eponential and Logarithmic Equations (Part I) In this section you will learn to: solve eponential equations using like ases solve eponential equations using logarithms solve logarithmic equations using

Return on Investment (ROI)

ROI 1 Return on Investment (ROI) Prepared by Sarah Major What is ROI? Return on investment (ROI) is a measure that investigates the amount of additional profits produced due to a certain investment. Businesses

ADDITION. Children should extend the carrying method to numbers with at least four digits.

Y5 AND Y6 ADDITION Children should extend the carrying method to numbers with at least four digits. 587 3587 + 475 + 675 1062 4262 1 1 1 1 1 Using similar methods, children will: add several numbers with

MATH-0910 Review Concepts (Haugen)

Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b,

9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

Exponential and Logarithmic Functions

Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,

Dr Brian Beaudrie pg. 1

Multiplication of Decimals Name: Multiplication of a decimal by a whole number can be represented by the repeated addition model. For example, 3 0.14 means add 0.14 three times, regroup, and simplify,

Accuplacer Arithmetic Study Guide

Accuplacer Arithmetic Study Guide Section One: Terms Numerator: The number on top of a fraction which tells how many parts you have. Denominator: The number on the bottom of a fraction which tells how

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA

FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x

Hands-On Math Algebra

Hands-On Math Algebra by Pam Meader and Judy Storer illustrated by Julie Mazur Contents To the Teacher... v Topic: Ratio and Proportion 1. Candy Promotion... 1 2. Estimating Wildlife Populations... 6 3.

Math 1050 Khan Academy Extra Credit Algebra Assignment

Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In

UNIT PLAN: EXPONENTIAL AND LOGARITHMIC FUNCTIONS

UNIT PLAN: EXPONENTIAL AND LOGARITHMIC FUNCTIONS Summary: This unit plan covers the basics of exponential and logarithmic functions in about 6 days of class. It is intended for an Algebra II class. The

FRACTIONS OPERATIONS

FRACTIONS OPERATIONS Summary 1. Elements of a fraction... 1. Equivalent fractions... 1. Simplification of a fraction... 4. Rules for adding and subtracting fractions... 5. Multiplication rule for two fractions...

Fractions and Linear Equations

Fractions and Linear Equations Fraction Operations While you can perform operations on fractions using the calculator, for this worksheet you must perform the operations by hand. You must show all steps

averages simple arithmetic average (arithmetic mean) 28 29 weighted average (weighted arithmetic mean) 32 33

537 A accumulated value 298 future value of a constant-growth annuity future value of a deferred annuity 409 future value of a general annuity due 371 future value of an ordinary general annuity 360 future

Substitute 4 for x in the function, Simplify.

Page 1 of 19 Review of Eponential and Logarithmic Functions An eponential function is a function in the form of f ( ) = for a fied ase, where > 0 and 1. is called the ase of the eponential function. The

Sample Problems. Practice Problems

Lecture Notes Quadratic Word Problems page 1 Sample Problems 1. The sum of two numbers is 31, their di erence is 41. Find these numbers.. The product of two numbers is 640. Their di erence is 1. Find these

No Solution Equations Let s look at the following equation: 2 +3=2 +7

5.4 Solving Equations with Infinite or No Solutions So far we have looked at equations where there is exactly one solution. It is possible to have more than solution in other types of equations that are

Absolute Value Equations and Inequalities

. Absolute Value Equations and Inequalities. OBJECTIVES 1. Solve an absolute value equation in one variable. Solve an absolute value inequality in one variable NOTE Technically we mean the distance between

Introduction to Fractions, Equivalent and Simplifying (1-2 days)

Introduction to Fractions, Equivalent and Simplifying (1-2 days) 1. Fraction 2. Numerator 3. Denominator 4. Equivalent 5. Simplest form Real World Examples: 1. Fractions in general, why and where we use

"Essential Mathematics & Statistics for Science" by Dr G Currell & Dr A A Dowman (John Wiley & Sons) Answers to In-Text Questions

"Essential Mathematics & Statistics for Science" by Dr G Currell & Dr A A Dowman (John Wiley & Sons) Answers to In-Text Questions 5 Logarithmic & Exponential Functions To navigate, use the Bookmarks in

Solutions to Exercises, Section 4.5

Instructor s Solutions Manual, Section 4.5 Exercise 1 Solutions to Exercises, Section 4.5 1. How much would an initial amount of \$2000, compounded continuously at 6% annual interest, become after 25 years?

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

Section 4.1 Rules of Exponents

Section 4.1 Rules of Exponents THE MEANING OF THE EXPONENT The exponent is an abbreviation for repeated multiplication. The repeated number is called a factor. x n means n factors of x. The exponent tells

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Section 9 Order of Operations

Supplemental Worksheet Problems To Accompany: The Pre-Algebra Tutor: Volume 1 Please watch Section 9 of this DVD before working these problems. The DVD is located at: http://www.mathtutordvd.com/products/item66.cfm

Solving DEs by Separation of Variables.

Solving DEs by Separation of Variables. Introduction and procedure Separation of variables allows us to solve differential equations of the form The steps to solving such DEs are as follows: dx = gx).

Unit 7: Radical Functions & Rational Exponents

Date Period Unit 7: Radical Functions & Rational Exponents DAY 0 TOPIC Roots and Radical Expressions Multiplying and Dividing Radical Expressions Binomial Radical Expressions Rational Exponents 4 Solving

HIBBING COMMUNITY COLLEGE COURSE OUTLINE

HIBBING COMMUNITY COLLEGE COURSE OUTLINE COURSE NUMBER & TITLE: - Beginning Algebra CREDITS: 4 (Lec 4 / Lab 0) PREREQUISITES: MATH 0920: Fundamental Mathematics with a grade of C or better, Placement Exam,

Fractions to decimals

Worksheet.4 Fractions and Decimals Section Fractions to decimals The most common method of converting fractions to decimals is to use a calculator. A fraction represents a division so is another way of

ECE 0142 Computer Organization. Lecture 3 Floating Point Representations

ECE 0142 Computer Organization Lecture 3 Floating Point Representations 1 Floating-point arithmetic We often incur floating-point programming. Floating point greatly simplifies working with large (e.g.,

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

Solving systems by elimination

December 1, 2008 Solving systems by elimination page 1 Solving systems by elimination Here is another method for solving a system of two equations. Sometimes this method is easier than either the graphing

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

Introduction to Macroeconomics TOPIC 2: The Goods Market

TOPIC 2: The Goods Market Annaïg Morin CBS - Department of Economics August 2013 Goods market Road map: 1. Demand for goods 1.1. Components 1.1.1. Consumption 1.1.2. Investment 1.1.3. Government spending

Chapter 7 Outline Math 236 Spring 2001

Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will

23. RATIONAL EXPONENTS

23. RATIONAL EXPONENTS renaming radicals rational numbers writing radicals with rational exponents When serious work needs to be done with radicals, they are usually changed to a name that uses exponents,

Activity 1: Using base ten blocks to model operations on decimals

Rational Numbers 9: Decimal Form of Rational Numbers Objectives To use base ten blocks to model operations on decimal numbers To review the algorithms for addition, subtraction, multiplication and division

Solutions of Linear Equations in One Variable

2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

Lecture Notes Order of Operations page 1

Lecture Notes Order of Operations page 1 The order of operations rule is an agreement among mathematicians, it simpli es notation. P stands for parentheses, E for exponents, M and D for multiplication

Math 120 Basic finance percent problems from prior courses (amount = % X base)

Math 120 Basic finance percent problems from prior courses (amount = % X base) 1) Given a sales tax rate of 8%, a) find the tax on an item priced at \$250, b) find the total amount due (which includes both

Multiplication and Division Properties of Radicals. b 1. 2. a Division property of radicals. 1 n ab 1ab2 1 n a 1 n b 1 n 1 n a 1 n b

488 Chapter 7 Radicals and Complex Numbers Objectives 1. Multiplication and Division Properties of Radicals 2. Simplifying Radicals by Using the Multiplication Property of Radicals 3. Simplifying Radicals

Simplification Problems to Prepare for Calculus

Simplification Problems to Prepare for Calculus In calculus, you will encounter some long epressions that will require strong factoring skills. This section is designed to help you develop those skills.

12) 13) 14) (5x)2/3. 16) x5/8 x3/8. 19) (r1/7 s1/7) 2

DMA 080 WORKSHEET # (8.-8.2) Name Find the square root. Assume that all variables represent positive real numbers. ) 6 2) 8 / 2) 9x8 ) -00 ) 8 27 2/ Use a calculator to approximate the square root to decimal