LOGARITHMS. Examples of Evaluating Logarithms by Using the Definition:
|
|
- Gregory Griffin
- 6 years ago
- Views:
Transcription
1 LOGARITHMS Definition of Logarithm: Given: a is a (real) number > 0 but a 1, called the base of the logarithm, and x is a (real) number > 0. log a x is defined to be the number y such that: y is the exponent for which a y = x. log a x = y ( log base number = exponent ) may also be read as follows: y is the logarithm to the base a of x OR, read another way: x is the antilogarithm to the base a of y. Examples of Evaluating Logarithms by Using the Definition: Example I: Find log log is the number/exponent y such that 10 y = 10,000. In order to find the answer, we start with the number 10 and multiply it by itself as follows: if y = 2, then 10 2 = = 100 if y = 3, then 10 3 = = 1000 if y = 4, then 10 4 = = 10,000. Therefore, 10 4 = 10,000 and we deduce that log = 4. Answer: log = 4. We read the answer as follows: 4 is the logarithm to the base 10 of the number Example II: Find log log = y is the exponent such that 5 y = After doing a little bit of calculation similar to Example I, we find that 5 5 = 3125, and, therefore, log = 5. Answer: log = 5. We read the answer as follows: 5 is the logarithm to the base 5 of the number 3125.
2 Example III: Find log 2 (1/4). log 2 (1/4) = y is the exponent such that 2 y = 1/4. Proceeding as in the previous examples, we obtain: if y = 1, then 2 1 = 2 if y = 2, then 2 2 = 2 2 = 4 if y = 3, then 2 3 = = 8 if y = 4, then 2 4 = = 16. Obviously, by choosing positive exponents, the numbers we obtained above were all greater than 1/4. Let s try 0 and some negative numbers as the exponent y instead. If y = 0, then 2 0 = 1. (From the Laws of Exponents, a 0 = 1 for any base a.) If y = -1, then 2 (-1) = 1 = 1/ If y = -2, then 2 (-2) = 1 = 1/ We have now found our answer. Therefore, log 2 (1/4) = -2. Answer: log 2 (1/4) = -2. Example IV: Find log log = y is the exponent y such that 253 y = 1. (From the Laws of Exponents, we know that a 0 = 1 for any base a.) Therefore, log = 0. Answer: log = 0. Example V: Find log If we proceed in a manner similar to the previous examples, we will conclude that log = 2 because (-7) 2 = 49. However, this answer is FALSE not because the arithmetic is incorrect, but because the logarithm function is NOT DEFINED for bases a 0. Answer: log is undefined. 2
3 Example VI: Find log 2 6. We want to find the exponent y such that 2 y = 6. From Example III, we know that 2 2 = 4 and 2 3 = 8. We can now surmise that the exponent y we are looking for is between 2 and 3, but, without a scientific calculator or a set of log tables, we will not be able to obtain the answer. Therefore, use your calculator to get the answer. Answer: log 2 6 = Examples of Evaluating Antilogarithms by Using the Definition: Example VII: Find the antilogarithm of 3 to the base 10. To find the antilog of 3 means that we are given y = 3 as the exponent and a = 10 as the base of the logarithm, and we are asked to find the number x such that log a x = y meaning that a y = x OR such that log 10 x = 3 meaning that 10 3 = x. Therefore, x = 10 3 = 1000 and we deduce that the antilogarithm of 3 to the base 10 is Answer: The antilog of 3 to the base 10 is Example VIII: Find the antilogarithm of 7 to the base 2. Without going through the detail, we can simply say that the antilogarithm of 7 to the base 2 is x = 2 7 = 128. Answer: The antilog of 7 to the base 2 is 128. From the above discussion, we observe that there seems to be an intimate connection between exponents and logarithms. In fact, there is a one-to-one correlation between the Laws of Exponents and the Properties of Logarithms. Laws of Exponents Properties of Logarithms a n a m = a (n + m) 1. log a (N M ) = log a N + log a M a n = a (n - m) 2. log a N = log a N - log a M a m M ( a n ) r n r = a 3. log a (N ) r = r log a N a 0 = 1 4. log a 1 = 0 a 1 = a 5. log a a = 1 3
4 Using the Properties of Logarithms to Solve Problems: Some hints as to how to recognize when to use the various properties of the logarithm. Use Property 1 if: number is a product of 2 factors Use Property 2 if: number is quotient of 2 numbers Use Property 3 if: number is a power Use Property 4 if: number is equal to 1 Use Property 5 if: number is equal to the base a Be aware that if the number is a combination of a product, a quotient, and a power, then various properties may need to be used in an appropriate sequence. ONLINE Logarithm Calculator: The following web site contains a Logarithm Calculator which will evaluate both logs and antilogarithms to any positive base a 1. Example 1: Simplify and evaluate log 80. If the base is not specified, it is assumed to be 10, which is the common base for the logarithm function. Therefore, the question becomes: Simplify and evaluate log Step 1: Simplify by trying to break down the size of the number 80 in some way and then use one of the properties of logarithms or the definition of logarithm to simplify further. We first note that 80 can be factored, keeping in mind that the base is 10. log = log = log log (by Property 1 as number is a product) = log (by Property 5) Step 2: Evaluate. Since the logarithm is to the base 10, use a calculator to evaluate. Therefore: log = log = = Answer: log = (rounded to 5 decimal places) 5
5 Example 2: Simplify and evaluate log 3 (17/6). HINT: Simplify by using property 2 (number is a quotient), followed by the change of base formula to base 10 (see page 75). Then evaluate using a calculator. log 10 (17/6) = log = log log 10 6 (by Property 2 as number 17/6 was a quotient) = (using your calculator) = Answer: log 10 (17/6) = (rounded to 5 decimal places) Example 3: Evaluate the number 2 25 using logs. Let x = Take logarithms of both sides of the equation. If the base is not specified in the problem, assume that the base is 10. (Actually you may use any base to solve this problem. The only glitch is that you have to have a way of calculating the logs and antilogarithms to the base that you choose.) Then log 10 x = log Because the number x is a power, we use Property 3 to obtain: log x = log = 25 log Evaluating log 10 2 by using your calculator, we deduce that the exponent to which the base 10 must be raised to get the number 2 is Therefore, log x = log = 25 log 10 2 = 25 ( ) = Because we originally wanted the value of x and we didn t want to multiply 2 by itself 25 times, we use the above equation containing the log of x and solve for x by taking the antilog to the base is 10 of both sides. Thus: x = antilog 10 (log 10 x) = antilog 10 ( ) (using your calculator) = e+7 (answer in scientific notation) = 33,554, (by changing to regular notation) = 33,554,432 (rounded to the nearest whole number) Answer: 2 25 = 33,554,432. Now, you may argue that the previous question could have done more easily and faster by multiplying 2 by itself 25 times. However, not all problems are more easily solved as a straight forward calculation. 6
6 Example 4: The formula for compound interest is the following: S = P (1+i) n where P = original investment, i = rate of interest, n = # of compounding periods, and S = value of investment at maturity. Suppose you want to invest $10000 and have $15000 after 3 years. Suppose also that your banker offers you semi-annual compounding on your investment. What annual interest rate would be necessary in order to fulfill your request? Is this feasible in today s financial market? In this problem, P = 10000, S = 15000, n = 6 (each year has 2 compounding periods since i is an annual interest rate) and you want to find i. Putting these values into the formula, we get: = (1+i) 6 and we need to solve for i. First, let us do some preliminary simplification. Recall that if we perform an operation on one side of the equation, we must perform the identical operation on the other side so that the equation is still valid. Therefore, by dividing both sides of the equation by and interchanging sides (so that the unknown is on the LHS), get: (1+i) 6 = 3/2. We could take the 6 th root of both sides to eliminate the power 6 on the LHS, but then we would have to have a way of finding the 6 th root of 3/2, which is not trivial or easily accessible. Therefore, let us resort to using logs. Taking the log of both sides, we get: log 10 (1+i) 6 = log 10 (3/2). By using Property 3 on the LHS and Property 2 on the RHS, we obtain: 6 log 10 (1+i) = log 10 (3)- log 10 (2) 6 log 10 (1+i) = (by evaluating the logs) 6 log 10 (1+i) = To get the factor containing i alone on the LHS, we divide both sides by 6 to get: log 10 (1+i) = Now we take antilogs of both sides. antilog 10 (log 10 (1+i)) = antilog 10 ( ) (1+i) = i = (subtract 1 from both sides) i = % (change decimal to a percentage) OR i = 7% per annum Answer: The annual interest rate necessary is 7% per annum. 7
LESSON ASSIGNMENT. After completing this lesson, you should be able to:
LESSON ASSIGNMENT LESSON 1 General Mathematics Review. TEXT ASSIGNMENT Paragraphs 1-1 through 1-49. LESSON OBJECTIVES After completing this lesson, you should be able to: 1-1. Identify and apply the properties
More information2.6 Logarithmic Functions. Inverse Functions. Question: What is the relationship between f(x) = x 2 and g(x) = x?
Inverse Functions Question: What is the relationship between f(x) = x 3 and g(x) = 3 x? Question: What is the relationship between f(x) = x 2 and g(x) = x? Definition (One-to-One Function) A function f
More informationAppendix A. Common Mathematical Operations in Chemistry
Appendix A Common Mathematical Operations in Chemistry In addition to basic arithmetic and algebra, four mathematical operations are used frequently in general chemistry: manipulating logarithms, using
More informationEXPONENTIAL AND LOGARITHMIC FUNCTIONS
Mathematics Revision Guides Exponential and Logarithmic Functions Page 1 of 14 M.K. HOME TUITION Mathematics Revision Guides Level: A-Level Year 1 / AS EXPONENTIAL AND LOGARITHMIC FUNCTIONS Version : 4.2
More informationLogarithms Tutorial for Chemistry Students
1 Logarithms 1.1 What is a logarithm? Logarithms Tutorial for Chemistry Students Logarithms are the mathematical function that is used to represent the number (y) to which a base integer (a) is raised
More informationSect Exponents: Multiplying and Dividing Common Bases
154 Sect 5.1 - Exponents: Multiplying and Dividing Common Bases Concept #1 Review of Exponential Notation In the exponential expression 4 5, 4 is called the base and 5 is called the exponent. This says
More informationLogarithms for analog circuits
Logarithms for analog circuits This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,
More informationLogarithms for analog circuits
Logarithms for analog circuits This worksheet and all related files are licensed under the Creative Commons Attribution License, version 1.0. To view a copy of this license, visit http://creativecommons.org/licenses/by/1.0/,
More informationAlgebra III: Blizzard Bag #1 Exponential and Logarithm Functions
NAME : DATE: PERIOD: Algebra III: Blizzard Bag #1 Exponential and Logarithm Functions Students need to complete the following assignment, which will aid in review for the end of course exam. Look back
More informationPart 4: Exponential and Logarithmic Functions
Part 4: Exponential and Logarithmic Functions Chapter 5 I. Exponential Functions (5.1) II. The Natural Exponential Function (5.2) III. Logarithmic Functions (5.3) IV. Properties of Logarithms (5.4) V.
More informationPre-Calc 2nd Semester Review Packet - #2
Pre-Calc 2nd Semester Review Packet - #2 Use the graph to determine the function's domain and range. 1) 2) Find the domain of the rational function. 3) h(x) = x + 8 x2-36 A) {x x -6, x 6, x -8} B) all
More informationPractice Questions for Final Exam - Math 1060Q - Fall 2014
Practice Questions for Final Exam - Math 1060Q - Fall 01 Before anyone asks, the final exam is cumulative. It will consist of about 50% problems on exponential and logarithmic functions, 5% problems on
More informationHonors Advanced Algebra Chapter 8 Exponential and Logarithmic Functions and Relations Target Goals
Honors Advanced Algebra Chapter 8 Exponential and Logarithmic Functions and Relations Target Goals By the end of this chapter, you should be able to Graph exponential growth functions. (8.1) Graph exponential
More informationThe trick is to multiply the numerator and denominator of the big fraction by the least common denominator of every little fraction.
Complex Fractions A complex fraction is an expression that features fractions within fractions. To simplify complex fractions, we only need to master one very simple method. Simplify 7 6 +3 8 4 3 4 The
More informationEssential Mathematics
Appendix B 1211 Appendix B Essential Mathematics Exponential Arithmetic Exponential notation is used to express very large and very small numbers as a product of two numbers. The first number of the product,
More informationExponential and Logarithmic Functions. Copyright Cengage Learning. All rights reserved.
Exponential and Logarithmic Functions Copyright Cengage Learning. All rights reserved. 4.5 Exponential and Logarithmic Equations Copyright Cengage Learning. All rights reserved. Objectives Exponential
More informationA. Incorrect! Check your algebra when you solved for volume. B. Incorrect! Check your algebra when you solved for volume.
AP Chemistry - Problem Drill 03: Basic Math for Chemistry No. 1 of 10 1. Unlike math problems, chemistry calculations have two key elements to consider in any number units and significant figures. Solve
More informationSection 4.7 Scientific Notation
Section 4.7 Scientific Notation INTRODUCTION Scientific notation means what it says: it is the notation used in many areas of science. It is used so that scientist and mathematicians can work relatively
More information2015 2nd Semester Exam Review
Algebra 2 2015 2nd Semester Exam Review 1. Write a function whose graph is a translation of the graph of the function in two directions. Describe the translation. 2. What are the solutions to the equation?
More informationAlgebra II. A2.1.1 Recognize and graph various types of functions, including polynomial, rational, and algebraic functions.
Standard 1: Relations and Functions Students graph relations and functions and find zeros. They use function notation and combine functions by composition. They interpret functions in given situations.
More informationChapter 2 Linear Equations and Inequalities in One Variable
Chapter 2 Linear Equations and Inequalities in One Variable Section 2.1: Linear Equations in One Variable Section 2.3: Solving Formulas Section 2.5: Linear Inequalities in One Variable Section 2.6: Compound
More informationIntermediate Algebra Chapter 12 Review
Intermediate Algebra Chapter 1 Review Set up a Table of Coordinates and graph the given functions. Find the y-intercept. Label at least three points on the graph. Your graph must have the correct shape.
More informationSection 3.7: Solving Radical Equations
Objective: Solve equations with radicals and check for extraneous solutions. In this section, we solve equations that have roots in the problem. As you might expect, to clear a root we can raise both sides
More informationMathematics Arithmetic Sequences
a place of mind F A C U L T Y O F E D U C A T I O N Department of Curriculum and Pedagogy Mathematics Arithmetic Sequences Science and Mathematics Education Research Group Supported by UBC Teaching and
More informationFinal Exam Study Guide Mathematical Thinking, Fall 2003
Final Exam Study Guide Mathematical Thinking, Fall 2003 Chapter R Chapter R contains a lot of basic definitions and notations that are used throughout the rest of the book. Most of you are probably comfortable
More informationSect 2.4 Multiplying and Dividing Integers
55 Sect 2.4 Multiplying and Dividing Integers Objective a: Understanding how to multiply two integers. To see how multiplying and dividing a negative and a positive number works, let s look at some examples.
More informationExponential Functions and Their Graphs (Section 3-1)
Exponential Functions and Their Graphs (Section 3-1) Essential Question: How do you graph an exponential function? Students will write a summary describing the steps for graphing an exponential function.
More informationCollege Algebra. Chapter 5 Review Created by: Lauren Atkinson. Math Coordinator, Mary Stangler Center for Academic Success
College Algebra Chapter 5 Review Created by: Lauren Atkinson Math Coordinator, Mary Stangler Center for Academic Success Note: This review is composed of questions from the chapter review at the end of
More informationMath 016 Lessons Wimayra LUY
Math 016 Lessons Wimayra LUY wluy@ccp.edu MATH 016 Lessons LESSON 1 Natural Numbers The set of natural numbers is given by N = {0, 1, 2, 3, 4...}. Natural numbers are used for two main reasons: 1. counting,
More informationEquations, Inequalities, and Problem Solving
CHAPTER Equations, Inequalities, and Problem Solving. Linear Equations in One Variable. An Introduction to Problem Solving. Formulas and Problem Solving.4 Linear Inequalities and Problem Solving Integrated
More informationEquations and Inequalities
Chapter 3 Equations and Inequalities Josef Leydold Bridging Course Mathematics WS 2018/19 3 Equations and Inequalities 1 / 61 Equation We get an equation by equating two terms. l.h.s. = r.h.s. Domain:
More informationReview questions for Math 111 final. Please SHOW your WORK to receive full credit Final Test is based on 150 points
Please SHOW your WORK to receive full credit Final Test is based on 150 points 1. True or False questions (17 pts) a. Common Logarithmic functions cross the y axis at (0,1) b. A square matrix has as many
More informationWarm Up Lesson Presentation Lesson Quiz. Holt McDougal Algebra 2
4-5 Warm Up Lesson Presentation Lesson Quiz Algebra 2 Warm Up Solve. 1. log 16 x = 3 2 64 2. log x 1.331 = 3 1.1 3. log10,000 = x 4 Objectives Solve exponential and logarithmic equations and equalities.
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, it is my epectation that you will have this packet completed. You will be way behind at the beginning of the year if you haven t attempted
More informationSection 2.3: Logarithmic Functions Lecture 3 MTH 124
Procedural Skills Learning Objectives 1. Build an exponential function using the correct compounding identifiers (annually, monthly, continuously etc...) 2. Manipulate exponents algebraically. e.g. Solving
More informationA factor times a logarithm can be re-written as the argument of the logarithm raised to the power of that factor
In this section we will be working with Properties of Logarithms in an attempt to take equations with more than one logarithm and condense them down into just a single logarithm. Properties of Logarithms:
More information13. [Place Value] units. The digit three places to the left of the decimal point is in the hundreds place. So 8 is in the hundreds column.
13 [Place Value] Skill 131 Understanding and finding the place value of a digit in a number (1) Compare the position of the digit to the position of the decimal point Hint: There is a decimal point which
More information9. TRANSFORMING TOOL #2 (the Multiplication Property of Equality)
9 TRANSFORMING TOOL # (the Multiplication Property of Equality) a second transforming tool THEOREM Multiplication Property of Equality In the previous section, we learned that adding/subtracting the same
More information31. TRANSFORMING TOOL #2 (the Multiplication Property of Equality)
3 TRANSFORMING TOOL # (the Multiplication Property of Equality) a second transforming tool THEOREM Multiplication Property of Equality In the previous section, we learned that adding/subtracting the same
More informationAlgebra II. Slide 1 / 261. Slide 2 / 261. Slide 3 / 261. Linear, Exponential and Logarithmic Functions. Table of Contents
Slide 1 / 261 Algebra II Slide 2 / 261 Linear, Exponential and 2015-04-21 www.njctl.org Table of Contents click on the topic to go to that section Slide 3 / 261 Linear Functions Exponential Functions Properties
More informationEquations. Rational Equations. Example. 2 x. a b c 2a. Examine each denominator to find values that would cause the denominator to equal zero
Solving Other Types of Equations Rational Equations Examine each denominator to find values that would cause the denominator to equal zero Multiply each term by the LCD or If two terms cross-multiply Solve,
More informationMATH Dr. Halimah Alshehri Dr. Halimah Alshehri
MATH 1101 haalshehri@ksu.edu.sa 1 Introduction To Number Systems First Section: Binary System Second Section: Octal Number System Third Section: Hexadecimal System 2 Binary System 3 Binary System The binary
More informationModeling with non-linear functions Business 8. Consider the supply curve. If we collect a few data points we might find a graph that looks like
Modeling with non-linear functions Business 8 Previously, we have discussed supply and demand curves. At that time we used linear functions. Linear models are often used when introducing concepts in other
More informationIn a previous lesson, we solved certain quadratic equations by taking the square root of both sides of the equation.
In a previous lesson, we solved certain quadratic equations by taking the square root of both sides of the equation. x = 36 (x 3) = 8 x = ± 36 x 3 = ± 8 x = ±6 x = 3 ± Taking the square root of both sides
More informationAlgebra II Chapter 5: Polynomials and Polynomial Functions Part 1
Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions
More informationChapter 2 Functions and Graphs
Chapter 2 Functions and Graphs Section 6 Logarithmic Functions Learning Objectives for Section 2.6 Logarithmic Functions The student will be able to use and apply inverse functions. The student will be
More informationReview Problems for the Final
Review Problems for the Final Math 0-08 008 These problems are provided to help you study The presence of a problem on this handout does not imply that there will be a similar problem on the test And the
More informationAlgebraic. techniques1
techniques Algebraic An electrician, a bank worker, a plumber and so on all have tools of their trade. Without these tools, and a good working knowledge of how to use them, it would be impossible for them
More informationSimplifying Radical Expressions
Simplifying Radical Expressions Product Property of Radicals For any real numbers a and b, and any integer n, n>1, 1. If n is even, then When a and b are both nonnegative. n ab n a n b 2. If n is odd,
More informationChapter 1A -- Real Numbers. iff. Math Symbols: Sets of Numbers
Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 1A! Page 1 Chapter 1A -- Real Numbers Math Symbols: iff or Example: Let A = {2, 4, 6, 8, 10, 12, 14, 16,...} and let B = {3, 6, 9, 12, 15, 18,
More informationSec. 4.2 Logarithmic Functions
Sec. 4.2 Logarithmic Functions The Logarithmic Function with Base a has domain all positive real numbers and is defined by Where and is the inverse function of So and Logarithms are inverses of Exponential
More informationEvaluate the exponential function at the specified value of x. 1) y = 4x, x = 3. 2) y = 2x, x = -3. 3) y = 243x, x = ) y = 16x, x = -0.
MAT 205-01C TEST 4 REVIEW (CHAP 13) NAME Evaluate the exponential function at the specified value of x. 1) y = 4x, x = 3 2) y = 2x, x = -3 3) y = 243x, x = 0.2 4) y = 16x, x = -0.25 Solve. 5) The number
More information2-7 Solving Absolute-Value Inequalities
Warm Up Solve each inequality and graph the solution. 1. x + 7 < 4 2. 14x 28 3. 5 + 2x > 1 When an inequality contains an absolute-value expression, it can be written as a compound inequality. The inequality
More informationp324 Section 5.2: The Natural Logarithmic Function: Integration
p324 Section 5.2: The Natural Logarithmic Function: Integration Theorem 5.5: Log Rule for Integration Let u be a differentiable function of x 1. 2. Example 1: Using the Log Rule for Integration ** Note:
More informationObjectives. Use the number e to write and graph exponential functions representing realworld
Objectives Use the number e to write and graph exponential functions representing realworld situations. Solve equations and problems involving e or natural logarithms. natural logarithm Vocabulary natural
More informationSample Problems. Lecture Notes Proof by Induction page 1. Prove each of the following statements by induction.
Lecture Notes Proof by Induction page Sample Problems Prove each of the following statements by induction.. For all natural numbers n; n (n + ) a) + + 3 + ::: + n. b) + + 3 + ::: + n n (n + ) (n + ). c)
More informationPrerequisite: Qualification by assessment process or completion of Mathematics 1050 or one year of high school algebra with a grade of "C" or higher.
Reviewed by: D. Jones Reviewed by: B. Jean Reviewed by: M. Martinez Text update: Spring 2017 Date reviewed: February 2014 C&GE Approved: March 10, 2014 Board Approved: April 9, 2014 Mathematics (MATH)
More informationSkills Practice Skills Practice for Lesson 4.1
Skills Practice Skills Practice for Lesson.1 Name Date Thinking About Numbers Counting Numbers, Whole Numbers, Integers, Rational and Irrational Numbers Vocabulary Define each term in your own words. 1.
More informationACCUPLACER MATH 0310
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to
More informationAPPENDIX B: Review of Basic Arithmetic
APPENDIX B: Review of Basic Arithmetic Personal Trainer Algebra Click Algebra in the Personal Trainer for an interactive review of these concepts. Equality = Is equal to 3 = 3 Three equals three. 3 = +3
More informationLesson 2: Introduction to Variables
In this lesson we begin our study of algebra by introducing the concept of a variable as an unknown or varying quantity in an algebraic expression. We then take a closer look at algebraic expressions to
More informationExponential and Logarithmic Equations
OpenStax-CNX module: m49366 1 Exponential and Logarithmic Equations OpenStax College This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 4.0 In this section,
More informationMAC Learning Objectives. Logarithmic Functions. Module 8 Logarithmic Functions
MAC 1140 Module 8 Logarithmic Functions Learning Objectives Upon completing this module, you should be able to 1. evaluate the common logarithmic function. 2. solve basic exponential and logarithmic equations.
More informationSection Summary. Sequences. Recurrence Relations. Summations Special Integer Sequences (optional)
Section 2.4 Section Summary Sequences. o Examples: Geometric Progression, Arithmetic Progression Recurrence Relations o Example: Fibonacci Sequence Summations Special Integer Sequences (optional) Sequences
More information3 Inequalities Absolute Values Inequalities and Intervals... 18
Contents 1 Real Numbers, Exponents, and Radicals 1.1 Rationalizing the Denominator................................... 1. Factoring Polynomials........................................ 1. Algebraic and Fractional
More informationChapter 7 - Exponents and Exponential Functions
Chapter 7 - Exponents and Exponential Functions 7-1: Multiplication Properties of Exponents 7-2: Division Properties of Exponents 7-3: Rational Exponents 7-4: Scientific Notation 7-5: Exponential Functions
More information5.2 Exponential and Logarithmic Functions in Finance
5. Exponential and Logarithmic Functions in Finance Question 1: How do you convert between the exponential and logarithmic forms of an equation? Question : How do you evaluate a logarithm? Question 3:
More information9.8 Exponential and Logarithmic Equations and Problem Solving
586 CHAPTER 9 Exponential and Logarithmic Functions 65. Find the amount of money Barbara Mack owes at the end of 4 years if 6% interest is compounded continuously on her $2000 debt. 66. Find the amount
More informationName Date Per. Ms. Williams/Mrs. Hertel
Name Date Per. Ms. Williams/Mrs. Hertel Day 7: Solving Exponential Word Problems involving Logarithms Warm Up Exponential growth occurs when a quantity increases by the same rate r in each period t. When
More informationTHE EXPONENTIAL AND NATURAL LOGARITHMIC FUNCTIONS: e x, ln x
Mathematics Revision Guides The Exponential and Natural Log Functions Page 1 of 17 M.K. HOME TUITION Mathematics Revision Guides Level: A-Level Year 1 / AS THE EXPONENTIAL AND NATURAL LOGARITHMIC FUNCTIONS:
More informationYOU CAN BACK SUBSTITUTE TO ANY OF THE PREVIOUS EQUATIONS
The two methods we will use to solve systems are substitution and elimination. Substitution was covered in the last lesson and elimination is covered in this lesson. Method of Elimination: 1. multiply
More informationAdlai E. Stevenson High School Course Description
Adlai E. Stevenson High School Course Description Content Objectives: Factor polynomial expressions, and solve, graph and write equations of quadratic functions. Graph and write equations of absolute value
More informationSolving Multi-Step Equations
1. Clear parentheses using the distributive property. 2. Combine like terms within each side of the equal sign. Solving Multi-Step Equations 3. Add/subtract terms to both sides of the equation to get the
More informationMathematics T (954) OVERALL PERFORMANCE RESPONSES OF CANDIDATES
Mathematics T (94) OVERALL PERFORMANCE The number of candidates for this subject was 7967. The percentage of the candidates who obtained a full pass was 68.09%, a increase of 2.6% when compared to the
More informationMath From Scratch Lesson 24: The Rational Numbers
Math From Scratch Lesson 24: The Rational Numbers W. Blaine Dowler May 23, 2012 Contents 1 Defining the Rational Numbers 1 1.1 Defining inverses........................... 2 1.2 Alternative Definition
More informationAlgebra 2 Honors: Final Exam Review
Name: Class: Date: Algebra 2 Honors: Final Exam Review Directions: You may write on this review packet. Remember that this packet is similar to the questions that you will have on your final exam. Attempt
More informationOrder of Operations. Real numbers
Order of Operations When simplifying algebraic expressions we use the following order: 1. Perform operations within a parenthesis. 2. Evaluate exponents. 3. Multiply and divide from left to right. 4. Add
More informationPractical Algebra. A Step-by-step Approach. Brought to you by Softmath, producers of Algebrator Software
Practical Algebra A Step-by-step Approach Brought to you by Softmath, producers of Algebrator Software 2 Algebra e-book Table of Contents Chapter 1 Algebraic expressions 5 1 Collecting... like terms 5
More informationAP Calculus AB Summer Assignment
AP Calculus AB Summer Assignment Name: When you come back to school, you will be epected to have attempted every problem. These skills are all different tools that you will pull out of your toolbo this
More informationWorksheet Topic 1 Order of operations, combining like terms 2 Solving linear equations 3 Finding slope between two points 4 Solving linear equations
Worksheet Topic 1 Order of operations, combining like terms 2 Solving linear equations 3 Finding slope between two points 4 Solving linear equations 5 Multiplying binomials 6 Practice with exponents 7
More informationPre Calculus Final Exam Review
Pre Calculus Final Exam Review Jun 2 10:04 PM Believe It or Not!! Jun 1 7:48 PM 1 Jun 2 9:58 PM Jun 2 10:06 PM 2 Jun 2 10:27 PM Jun 1 7:55 PM 3 Jun 2 10:20 PM Jun 2 10:22 PM 4 Jun 2 10:22 PM Jun 3 9:43
More informationNotes on arithmetic. 1. Representation in base B
Notes on arithmetic The Babylonians that is to say, the people that inhabited what is now southern Iraq for reasons not entirely clear to us, ued base 60 in scientific calculation. This offers us an excuse
More informationnt and A = Pe rt to solve. 3) Find the accumulated value of an investment of $10,000 at 4% compounded semiannually for 5 years.
Exam 4 Review Approximate the number using a calculator. Round your answer to three decimal places. 1) 2 1.7 2) e -1.4 Use the compound interest formulas A = P 1 + r n nt and A = Pe rt to solve. 3) Find
More informationFinal Exam Review: Study Guide Math 3
Final Exam Review: Study Guide Math 3 Name: Day 1 Functions, Graphing, Regression Relation: Function: Domain: Range: Asymptote: Hole: Graphs of Functions f(x) = x f(x) = f(x) = x f(x) = x 3 Key Ideas Key
More informationArithmetic Testing OnLine (ATOL) SM Assessment Framework
Arithmetic Testing OnLine (ATOL) SM Assessment Framework Overview Assessment Objectives (AOs) are used to describe the arithmetic knowledge and skills that should be mastered by the end of each year in
More informationExponential and Logarithmic Functions
Graduate T.A. Department of Mathematics Dynamical Systems and Chaos San Diego State University April 9, 11 Definition (Exponential Function) An exponential function with base a is a function of the form
More informationProperties of Logarithms
Warm Up Simplify. 1. (2 6 )(2 8 ) 2 14 2. (3 2 )(3 5 ) 3 3 3 8 3. 4. 4 4 5. (7 3 ) 5 7 15 Write in exponential form. 6. log x x = 1 x 1 = x 7. 0 = log x 1 x 0 = 1 Objectives Use properties to simplify
More informationAssume that you have made n different measurements of a quantity x. Usually the results of these measurements will vary; call them x 1
#1 $ http://www.physics.fsu.edu/users/ng/courses/phy2048c/lab/appendixi/app1.htm Appendix I: Estimates for the Reliability of Measurements In any measurement there is always some error or uncertainty in
More informationAlgebra 1. Math Review Packet. Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals
Algebra 1 Math Review Packet Equations, Inequalities, Linear Functions, Linear Systems, Exponents, Polynomials, Factoring, Quadratics, Radicals 2017 Math in the Middle 1. Clear parentheses using the distributive
More informationExponential Functions Concept Summary See pages Vocabulary and Concept Check.
Vocabulary and Concept Check Change of Base Formula (p. 548) common logarithm (p. 547) exponential decay (p. 524) exponential equation (p. 526) exponential function (p. 524) exponential growth (p. 524)
More information1. The dosage in milligrams D of a heartworm preventive for a dog who weighs X pounds is given by D x. Substitute 28 in place of x to get:
1. The dosage in milligrams D of a heartworm preventive for a dog who weighs X pounds is given by D x 28 pounds. ( ) = 136 ( ). Find the proper dosage for a dog that weighs 25 x Substitute 28 in place
More informationA. Incorrect! Replacing is not a method for solving systems of equations.
ACT Math and Science - Problem Drill 20: Systems of Equations No. 1 of 10 1. What methods were presented to solve systems of equations? (A) Graphing, replacing, and substitution. (B) Solving, replacing,
More informationGeneral Mathematics Paper 2,May/June
General Mathematics Paper 2,May/June. 2007 Question 1 (a). Evaluate, without using mathematical tables or calculator, (3.69 x 105) / (1.64 x 10-3), leaving your answer in the standard form. (b). A man
More informationMATH 9 YEAR END REVIEW
Name: Block: MATH 9 YEAR END REVIEW Complete the following reviews in pencil. Use a separate piece of paper if you need more space. Please pay attention to whether you can use a calculator or not for each
More informationPolynomials and Rational Functions (2.1) The shape of the graph of a polynomial function is related to the degree of the polynomial
Polynomials and Rational Functions (2.1) The shape of the graph of a polynomial function is related to the degree of the polynomial Shapes of Polynomials Look at the shape of the odd degree polynomials
More information8-4. Negative Exponents. What Is the Value of a Power with a Negative Exponent? Lesson. Negative Exponent Property
Lesson 8-4 Negative Exponents BIG IDEA The numbers x n and x n are reciprocals. What Is the Value of a Power with a Negative Exponent? You have used base 10 with a negative exponent to represent small
More informationDefinition of a Logarithm
Chapter 17 Logarithms Sec. 1 Definition of a Logarithm In the last chapter we solved and graphed exponential equations. The strategy we used to solve those was to make the bases the same, set the exponents
More informationWe will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.).
College algebra We will review simplifying radicals, exponents and their rules, multiplying polynomials, factoring polynomials, greatest common denominators, and solving rational equations. Pre-requisite
More informationGRE Quantitative Reasoning Practice Questions
GRE Quantitative Reasoning Practice Questions y O x 7. The figure above shows the graph of the function f in the xy-plane. What is the value of f (f( ))? A B C 0 D E Explanation Note that to find f (f(
More informationDecimal Addition: Remember to line up the decimals before adding. Bring the decimal straight down in your answer.
Summer Packet th into 6 th grade Name Addition Find the sum of the two numbers in each problem. Show all work.. 62 2. 20. 726 + + 2 + 26 + 6 6 Decimal Addition: Remember to line up the decimals before
More information