Section 1.5 An Introduction to Logarithms
|
|
- Christiana Walters
- 5 years ago
- Views:
Transcription
1 Section. An Introduction to Logarithms So far we ve used the idea exponent Base Result from two points of view. When the base and exponent were given, for instance, we simplified to the result 8. When the exponent and result were given, for instance 8, we simplified to the base. A logarithm asks us to use the same idea from a third point of view. With a logarithm we re given a base and result and our job will be to find the exponent. In this section I ll introduce logarithms and show the relationship between logarithmic and exponential form... Some Vocabulary for Logarithms. A logarithm is the exponent a base must be raised to, to produce a result. The logarithm symbol, log, implies we are interested in finding the exponent. The original base, known as the base of the logarithm, is written a little lower and a little smaller log base. The original result is written to the right and is often called the argument log Argument. For example the expression log 8 is asking you to start with a base base of and find an exponent that returns a result of 8 so log 8. Definition: Logarithm English: The exponent a base must be raised to, to get a given result. Example: log 8 Note: The base should be greater than 0 and not equal to... Logarithmic and Exponential Form After finding the logarithm you can rewrite your work in exponential form to verify your answer. Definition: The Logarithm Form, Exponential Form Equivalence English: A logarithmic relationship can be written in exponential form. Example: log For instance earlier we found that log 8. Rewriting this equation in exponential form as 8 can help verify the answer was correct. Practice.. Logarithmic and Exponential Form a) log log First state the question the expression is asking. Then simplify the expression. Finally, rewrite your work in exponential form and simplify to check. log What exponent of two gives a result of sixteen? log Simplified the expression. Rewrote in exponential form to verify the work. log Five to what power gives a result of five? log Simplified the expression Rewrote in exponential form to verify the work. Copyright 0 Scott Storla. An Introduction to Logarithms
2 Homework. First state the question the expression is asking. Then simplify the expression. Finally, rewrite your work in exponential form and simplify to check. ) log ) log 000 ) 0 Log 8 ) log ) log ) log.. The Common and Natural Logarithm A scientific calculator has two different keys for logarithms. The common log key log implies the base is 0 so log is log. Base 0 is useful because our number system is a base 0 system. The natural 0 log key ln is used when the base is e so ln is log e. Recall that e is an irrational constant with a value approximately equal to.8. If you re lucky enough to study a first course in calculus you may learn why the name natural was chosen when discussing logarithms with a base of e. There s no agreement by calculator manufacturers on the order of the buttons you push to find log or ln. On some calculators you put in your argument and then push your log or ln key. On others you push the log or ln key first and then put in the argument. Try finding log00. Your calculator should say. Now try to find ln00. Your calculator should say.0 if you round to the ten-thousandths place. To check your work using exponential form you ll often use the power key or the if log00 use the power key to see if0 00. To check if ln00.0 use the x e key. To check x e key to see if.0 e 00. The wavy equals sign is necessary because.0 is only approximately equal to the real exponent (remember we rounded to the thousandths place). Since e , which is very close to 00, you should be comfortable with the answer. Practice.. The Common and Natural Logarithm a) log ln Simplify using your calculator. Rewrite your work in exponential form and simplify to check your work The approximate answer from the calculator Rewrote in exponential form and used the calculator to check the work. 0.9 Rounded the answer. e Checking the exponential form confirms the work. Homework. Simplify using your calculator. Check your answer. ) ln 8) log 9) log 0) ln ) log 9 ) ln. In some books log stands for log e. It won t in this course, but it s something you should be aware of. Copyright 0 Scott Storla. An Introduction to Logarithms
3 .. The Change of Base Formula Often calculators only have keys for log or ln. If your original base isn t 0 or e you can still use your calculator as long as you know the change of base formula. Procedure - The Change of Base Formula Using ln. To change a logarithmic expression to an equivalent expression with base e, build a fraction. The numerator is the natural log of the original argument. The denominator is the natural log of the original base. Example: log ln9 9 ln Practice.. The Change of Base Formula Simplify using the change of base formula. When appropriate round to the tenthousandths place. Use exponential form to check your answer. a) log 9 ln9 ln Used the change of base formula with base e. Simplified. Using log9 log would have given the same answer. log 00 log00 log Used the change of base formula with base 0..9 Simplified using a calculator. Rounded to the ten-thousandths place Checked the answer. Homework. Simplify using the change of base formula. When appropriate round to the tenthousandths place. Use exponential form to check your answer. ) log 0.0 ) log 8) ) 9 log ) log,000 log. ) 8 log Logarithms and the Order of Operations Simplifying logarithms is included in line of the order of operations. Also, the argument of a logarithm is a type of implicit grouping. Practice.. Logarithms and the Order of Operations a) ln8,000 /,000 Count the number of operations, name the operations using the correct order and then simplify the expression. If necessary round to the thousandths place. ln(.) Divided. There are two operations. The division would be first and then taking the natural logarithm. 0. Took the natural logarithm. Copyright 0 Scott Storla. An Introduction to Logarithms
4 log. There are three operations. Taking log base 0 of fifteen would be first, then multiplication by and finally division by Took log of and rounded. 0. Multiplied by and divided the result by. Homework. Count the number of operations, name the operations using the correct order and then simplify the expression. Only use a calculator when necessary. 9) ln 0) log9 ) ln 80 0 ) 00,000 ) ln ,000 log0 log0 ) ln 8, ,000 ) ln... ) ln 0. 8) ) ln ln ln 0,000 ln,00 9) 0 ln 0 0) ln ln ln Copyright 0 Scott Storla. An Introduction to Logarithms
5 Homework. ) Six to what power gives a result of thirty-six?,, ) What exponent of ten gives a result of one thousand?,, ) Three to what power results in eighty-one?,, ) What power of four gives a result of four?,, 8 ) What exponent of six results in two hundred sixteen?,, ) Two to what power gives a result of thirty-two?,, 0,000 ) 0. 8).09 9).8 0) 0 ) 0. ) 0.8 ).9 ) ) 0.08 ) ) 8).8 9) There are two operations. First multiply and then take the logarithm. The answer is about.0. 0) There are two operations. First add then take the logarithm. The answer is about.9. ) There are five operations. First square the base of 0, next find the logarithms left to right, then multiply by and add last. The answer is. ) There are four operations. The division is first, the logarithm is next, followed by the exponent of negative one and then the multiplication. The answer is 9.. ) There are three operations. The division inside the parentheses is first, the logarithm is next and the division is last. The answer is about ) There are four operations. The division is first, the logarithm is next, followed by the exponent of negative one and then the multiplication. The answer is about.. ) There are three operations. First simplify the argument by multiplying and then subtracting. Then take the logarithm. The answer is about ) There are four operations. The division is first, the logarithm is next, followed by the exponent of negative one and then the multiplication. The answer is about.0. ) There are six operations. The division in the argument is first and then the logarithms and subtraction inside the parentheses. The logarithm outside the parentheses follows and the subtraction is last. The answer is 0. 8) There are three operations. The division inside the parentheses is first, the logarithm is next and the division is last. The answer is about ) There are five operations. First subtract and divide to simplify the argument. Then take the logarithm and multiply the quotient of one-tenth to the result. The answer is about ) There are five operations. First take the natural logarithms left to right and then add and subtract left to right. The answer is 0. Copyright 0 Scott Storla. An Introduction to Logarithms
Exponential and Logarithmic Functions. Copyright Cengage Learning. All rights reserved.
5 Exponential and Logarithmic Functions Copyright Cengage Learning. All rights reserved. 5.3 Properties of Logarithms Copyright Cengage Learning. All rights reserved. Objectives Use the change-of-base
More informationLogarithms. Since perhaps it s been a while, calculate a few logarithms just to warm up.
Logarithms Since perhaps it s been a while, calculate a few logarithms just to warm up. 1. Calculate the following. (a) log 3 (27) = (b) log 9 (27) = (c) log 3 ( 1 9 ) = (d) ln(e 3 ) = (e) log( 100) =
More informationRadical Expressions and Graph (7.1) EXAMPLE #1: EXAMPLE #2: EXAMPLE #3: Find roots of numbers (Objective #1) Figure #1:
Radical Expressions and Graph (7.1) Find roots of numbers EXAMPLE #1: Figure #1: Find principal (positive) roots EXAMPLE #2: Find n th roots of n th powers (Objective #3) EXAMPLE #3: Figure #2: 7.1 Radical
More informationYou could identify a point on the graph of a function as (x,y) or (x, f(x)). You may have only one function value for each x number.
Function Before we review exponential and logarithmic functions, let's review the definition of a function and the graph of a function. A function is just a rule. The rule links one number to a second
More informationComparing Exponential and Logarithmic Rules
Name _ Date Period Comparing Exponential and Logarithmic Rules Task : Looking closely at exponential and logarithmic patterns ) In a prior lesson you graphed and then compared an exponential function with
More informationPre-Algebra Unit 1: Number Sense Unit 1 Review Packet
Pre-Algebra Unit 1: Number Sense Unit 1 Review Packet Target 1: Writing Repeating Decimals in Rational Form Remember the goal is to get rid of the repeating decimal so we can write the number in rational
More informationLogarithms ID1050 Quantitative & Qualitative Reasoning
Logarithms ID1050 Quantitative & Qualitative Reasoning History and Uses We noticed that when we multiply two numbers that are the same base raised to different exponents, that the result is the base raised
More information171S5.4p Properties of Logarithmic Functions. November 20, CHAPTER 5: Exponential and Logarithmic Functions. Examples. Express as a product.
MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 5: Exponential and Logarithmic Functions 5.1 Inverse Functions 5.2 Exponential Functions and Graphs 5.3 Logarithmic Functions
More informationProperties of Logarithms
Properties of Logarithms Accelerated Pre-Calculus Mr. Niedert Accelerated Pre-Calculus Properties of Logarithms Mr. Niedert 1 / 14 Properties of Logarithms 1 Change-of-Base Formula Accelerated Pre-Calculus
More informationNOTES: SIGNED INTEGERS DAY 1
NOTES: SIGNED INTEGERS DAY 1 MULTIPLYING and DIVIDING: Same Signs (POSITIVE) + + = + positive x positive = positive = + negative x negative = positive Different Signs (NEGATIVE) + = positive x negative
More informationProperties of Logarithms
Properties of Logarithms Warm Up Lesson Presentation Lesson Quiz Algebra 2 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
More informationMA10103: Foundation Mathematics I. Lecture Notes Week 3
MA10103: Foundation Mathematics I Lecture Notes Week 3 Indices/Powers In an expression a n, a is called the base and n is called the index or power or exponent. Multiplication/Division of Powers a 3 a
More informationINTRODUCTION TO LOGARITHMS
INTRODUCTION TO LOGARITHMS Dear Reader Logarithms are a tool originally designed to simplify complicated arithmetic calculations. They were etensively used before the advent of calculators. Logarithms
More informationSection 7.2 Logarithmic Functions
Math 150 c Lynch 1 of 6 Section 7.2 Logarithmic Functions Definition. Let a be any positive number not equal to 1. The logarithm of x to the base a is y if and only if a y = x. The number y is denoted
More informationFocus on Mathematics
Focus on Mathematics Year 4 Pre-Learning Tasks Number Pre-learning tasks are used at the start of each new topic in Maths. The children are grouped after the pre-learning task is marked to ensure the work
More informationMath Fundamentals for Statistics (Math 52) Unit 2:Number Line and Ordering. By Scott Fallstrom and Brent Pickett The How and Whys Guys.
Math Fundamentals for Statistics (Math 52) Unit 2:Number Line and Ordering By Scott Fallstrom and Brent Pickett The How and Whys Guys Unit 2 Page 1 2.1: Place Values We just looked at graphing ordered
More informationLesson 1 6. Algebra: Variables and Expression. Students will be able to evaluate algebraic expressions.
Lesson 1 6 Algebra: Variables and Expression Students will be able to evaluate algebraic expressions. P1 Represent and analyze patterns, rules and functions with words, tables, graphs and simple variable
More informationExtra Practice 1. Name Date. Lesson 1: Numbers in the Media. 1. Rewrite each number in standard form. a) 3.6 million b) 6 billion c)
Master 4.27 Extra Practice 1 Lesson 1: Numbers in the Media 1. Rewrite each number in standard form. 3 a) 3.6 million b) 6 billion c) 1 million 4 2 1 d) 2 billion e) 4.25 million f) 1.4 billion 10 2. Use
More informationExtra Practice 1. Name Date. Lesson 1: Numbers in the Media. 1. Rewrite each number in standard form. a) 3.6 million
Master 4.27 Extra Practice 1 Lesson 1: Numbers in the Media 1. Rewrite each number in standard form. a) 3.6 million 3 b) 6 billion 4 c) 1 million 2 1 d) 2 billion 10 e) 4.25 million f) 1.4 billion 2. Use
More informationDecember 10, Unit Two - Operations with decimals
Unit Two - Operations with decimals Unit Two - Operations with Decimals Introduction (or re-introduction) to place value Read the following numbers (properly!) 145 2.35 1 567 043.793 Place Value 1,000,000
More informationThe bottom number in the fraction is called the denominator. The top number is called the numerator.
For Topics 8 and 9, the students should know: Fractions are a part of a whole. The bottom number in the fraction is called the denominator. The top number is called the numerator. Equivalent fractions
More information18 Logarithmic Functions
18 Logarithmic Functions Concepts: Logarithms (Section 3.3) Logarithms as Functions Logarithms as Exponent Pickers Inverse Relationship between Logarithmic and Exponential Functions. The Common Logarithm
More informationMathematics for Biology
MAT1142 Department of Mathematics University of Ruhuna A.W.L. Pubudu Thilan Logarithms Why do we need logarithms? Sometimes you only care about how big a number is relative to other numbers. The Richter,
More informationBy Scott Fallstrom and Brent Pickett The How and Whys Guys
Math Fundamentals for Statistics I (Math 52) Unit 2:Number Line and Ordering By Scott Fallstrom and Brent Pickett The How and Whys Guys This work is licensed under a Creative Commons Attribution- NonCommercial-ShareAlike
More informationJ.7 Properties of Logarithms
J.7. PROPERTIES OF LOGARITHMS 1 J.7 Properties of Logarithms J.7.1 Understanding Properties of Logarithms Product Rule of Logarithms log a MN = log a M +log a N Example J.7.1. Rewrite as a sum of logarithms:
More informationHW#02 (18 pts): All recommended exercises from JIT (1 pt/problem)
Spring 2011 MthSc103 Course Calendar Page 1 of 7 January W 12 Syllabus/Course Policies BST Review Th 13 Basic Skills Test F 14 JIT 1.1 1.3: Numbers, Fractions, Parentheses JIT 1.1: 2, 6, 8, 9 JIT 1.2:
More informationPowers and roots 6.1. Previous learning. Objectives based on NC levels and (mainly level ) Lessons 1 Squares, cubes and roots.
N 6.1 Powers and roots Previous learning Before they start, pupils should be able to: use index notation and the index laws for positive integer powers understand and use the order of operations, including
More informationSchool of Business. Blank Page
Logarithm The purpose of this unit is to equip the learners with the concept of logarithm. Under the logarithm, the topics covered are nature of logarithm, laws of logarithm, change the base of logarithm,
More informationWorking with Integer Exponents
4.2 Working with Integer Exponents GOAL Investigate powers that have integer or zero exponents. LEARN ABOUT the Math The metric system of measurement is used in most of the world. A key feature of the
More informationCore Connections, Course 2 Checkpoint Materials
Core Connections, Course Checkpoint Materials Notes to Students (and their Teachers) Students master different skills at different speeds. No two students learn exactly the same way at the same time. At
More informationWheels Diameter / Conversion of Units
Note to the teacher On this page, students will learn about the relationships between wheel diameter, circumference, revolutions and distance. They will also convert measurement units and use fractions
More informationa) 1/2 b) 3/7 c) 5/8 d) 4/10 e) 5/15 f) 2/4 a) two-fifths b) three-eighths c) one-tenth d) two-thirds a) 6/7 b) 7/10 c) 5/50 d) ½ e) 8/15 f) 3/4
MATH M010 Unit 2, Answers Section 2.1 Page 72 Practice 1 a) 1/2 b) 3/7 c) 5/8 d) 4/10 e) 5/15 f) 2/4 Page 73 Practice 2 a) two-fifths b) three-eighths c) one-tenth d) two-thirds e) four-ninths f) one quarter
More informationPREREQUISITE/PRE-CALCULUS REVIEW
PREREQUISITE/PRE-CALCULUS REVIEW Introduction This review sheet is a summary of most of the main topics that you should already be familiar with from your pre-calculus and trigonometry course(s), and which
More informationCopyright 2009 Pearson Canada Inc., Toronto, Ontario.
Copyright 2009 Pearson Canada Inc., Toronto, Ontario. All rights reserved. This publication (work) is protected by copyright. You are authorized to print one copy of this publication (work) for your personal,
More informationOutcome 9 Review Foundations and Pre-Calculus 10
Outcome 9 Review Foundations and Pre-Calculus 10 Level 2 Example: Writing an equation in slope intercept form Slope-Intercept Form: y = mx + b m = slope b = y-intercept Ex : Write the equation of a line
More informationLogarithmic Functions and Their Graphs
Logarithmic Functions and Their Graphs Accelerated Pre-Calculus Mr. Niedert Accelerated Pre-Calculus Logarithmic Functions and Their Graphs Mr. Niedert 1 / 24 Logarithmic Functions and Their Graphs 1 Logarithmic
More informationLogs and Exponentials Higher.notebook February 26, Daily Practice
Daily Practice 2.2.2015 Daily Practice 3.2.2015 Today we will be learning about exponential functions and logs. Homework due! Need to know for Unit Test 2: Expressions and Functions Adding and subtracng
More informationLogarithmic Functions
C H A P T ER Logarithmic Functions The human ear is capable of hearing sounds across a wide dynamic range. The softest noise the average human can hear is 0 decibels (db), which is equivalent to a mosquito
More informationMath 081 Exam 1 Preparation V01 Ch 1-2 Winter 2010 Winter 2010 Dressler NO CALCULATOR/NO NOTES/NO BOOK/55 MINUTES. Name
Math 081 Exam 1 Preparation V01 Ch 1-2 Winter 2010 Winter 2010 Dressler NO CALCULATOR/NO NOTES/NO BOOK/55 MINUTES Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers
More informationNumber Sense and Decimal Unit Notes
Number Sense and Decimal Unit Notes Table of Contents: Topic Page Place Value 2 Rounding Numbers 2 Face Value, Place Value, Total Value 3 Standard and Expanded Form 3 Factors 4 Prime and Composite Numbers
More informationMental Calculation Policy 2014
Mental Calculation Policy 2014 RECEPTION Children count reliably with numbers from one to 20 and place them in order. Children can say which number is one more or one less than a given number up to 20
More informationSolving Equations and Graphing
Solving Equations and Graphing Question 1: How do you solve a linear equation? Answer 1: 1. Remove any parentheses or other grouping symbols (if necessary). 2. If the equation contains a fraction, multiply
More informationName: Class: Date: Class Notes - Division Lesson Six. 1) Bring the decimal point straight up to the roof of the division symbol.
Name: Class: Date: Goals:11 1) Divide a Decimal by a Whole Number 2) Multiply and Divide by Powers of Ten 3) Divide by Decimals To divide a decimal by a whole number: Class Notes - Division Lesson Six
More informationStation 1. Rewrite each number using Scientific Notation 1. 6,890,000 = ,560,000 = 3. 1,500,000,000 = 4. 8,200 = 6. 0.
Station 1 Rewrite each number using Scientific Notation 1. 6,890,000 = 2. 240,560,000 = 3. 1,500,000,000 = 4. 8,200 = 5. 50 = 6. 0.00000000265 = 7. 0.0009804 = 8. 0.000080004 = 9. 0.5 = Station 2 Add using
More information3.3 Properties of Logarithms
Section 3.3 Properties of Logarithms 07 3.3 Properties of Logarithms Change of Base Most calculators have only two types of log keys, one for common logarithms (base 0) and one for natural logarithms (base
More informationChapter 0 Getting Started on the TI-83 or TI-84 Family of Graphing Calculators
Chapter 0 Getting Started on the TI-83 or TI-84 Family of Graphing Calculators 0.1 Turn the Calculator ON / OFF, Locating the keys Turn your calculator on by using the ON key, located in the lower left
More informationLesson 0.1 The Same yet Smaller
Lesson 0.1 The Same yet Smaller 1. Write an expression and find the total shaded area in each square. In each case, assume that the area of the largest square is 1. a. b. c. d. 2. Write an expression and
More information2.5. Combining Powers. LEARN ABOUT the Math. Nicole and Yvonne made origami paper cubes for a math project.
2.5 Combining Powers YOU WILL NEED a calculator Nicole s cube 4 cm GOAL Simplify products and quotients of powers with the same exponent. LEARN ABOUT the Math Nicole and Yvonne made origami paper cubes
More informationChapter 3 Exponential and Logarithmic Functions
Chapter 3 Exponential and Logarithmic Functions Section 1 Section 2 Section 3 Section 4 Section 5 Exponential Functions and Their Graphs Logarithmic Functions and Their Graphs Properties of Logarithms
More informationObjectives: Students will learn to divide decimals with both paper and pencil as well as with the use of a calculator.
Unit 3.5: Fractions, Decimals and Percent Lesson: Dividing Decimals Objectives: Students will learn to divide decimals with both paper and pencil as well as with the use of a calculator. Procedure: Dividing
More informationfind more or less than a given number find 10 or 100 more or less than a given number
count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number Number: Number and Place Value COUNTING Consolidate count to and across 100, forwards and backwards, beginning
More informationMath 147 Section 5.2. Application Example
Math 147 Section 5.2 Logarithmic Functions Properties of Change of Base Formulas Math 147, Section 5.2 1 Application Example Use a change-of-base formula to evaluate each logarithm. (a) log 3 12 (b) log
More informationExponential equations: Any equation with a variable used as part of an exponent.
Write the 4 steps for solving Exponential equations Exponential equations: Any equation with a variable used as part of an exponent. OR 1) Make sure one and only one side of the equation is only a base
More information1 Integers and powers
1 Integers and powers 1.1 Integers and place value An integer is any positive or negative whole number. Zero is also an integer. The value of a digit in a number depends on its position in the number.
More informationMath 104: Homework Exercises
Math 04: Homework Exercises Chapter 5: Decimals Ishibashi Chabot College Fall 20 5. Reading and Writing Decimals In the number 92.7845, identify the place value of the indicated digit.. 8 2.. 4. 7 Write
More informationMath Lecture 2 Inverse Functions & Logarithms
Math 1060 Lecture 2 Inverse Functions & Logarithms Outline Summary of last lecture Inverse Functions Domain, codomain, and range One-to-one functions Inverse functions Inverse trig functions Logarithms
More informationInstructor Notes for Chapter 4
Section 4.1 One to One Functions (Day 1) Instructor Notes for Chapter 4 Understand that an inverse relation undoes the original Understand why the line y = xis a line of symmetry for the graphs of relations
More informationL_sson 9 Subtracting across zeros
L_sson 9 Subtracting across zeros A. Here are the steps for subtracting 3-digit numbers across zeros. Complete the example. 7 10 12 8 0 2 2 3 8 9 1. Subtract the ones column. 2 8 requires regrouping. 2.
More informationAlex Benn. Math 7 - Outline First Semester ( ) (Numbers in parentheses are the relevant California Math Textbook Sections) Quarter 1 44 days
Math 7 - Outline First Semester (2016-2017) Alex Benn (Numbers in parentheses are the relevant California Math Textbook Sections) Quarter 1 44 days 0.1 Classroom Rules Multiplication Table Unit 1 Measuring
More informationCH 20 NUMBER WORD PROBLEMS
187 CH 20 NUMBER WORD PROBLEMS Terminology To double a number means to multiply it by 2. When n is doubled, it becomes 2n. The double of 12 is 2(12) = 24. To square a number means to multiply it by itself.
More informationCorrelation of USA Daily Math Grade 5 to Common Core State Standards for Mathematics
Correlation of USA Daily Math Grade 5 to Common Core State Standards for Mathematics 5.OA Operations and Algebraic Thinking (Mondays) 5.OA.1 Use parentheses, brackets, or p. 1 #3 p. 7 #3 p. 12 Brain Stretch
More information8.1 Exponential Growth 1. Graph exponential growth functions. 2. Use exponential growth functions to model real life situations.
8.1 Exponential Growth Objective 1. Graph exponential growth functions. 2. Use exponential growth functions to model real life situations. Key Terms Exponential Function Asymptote Exponential Growth Function
More information4 What are and 31,100-19,876? (Two-part answer)
1 What is 14+22? 2 What is 68-37? 3 What is 14+27+62+108? 4 What are 911-289 and 31,100-19,876? (Two-part answer) 5 What are 4 6, 7 8, and 12 5? (Three-part answer) 6 How many inches are in 4 feet? 7 How
More information+ 4 ~ You divided 24 by 6 which equals x = 41. 5th Grade Math Notes. **Hint: Zero can NEVER be a denominator.**
Basic Fraction numerator - (the # of pieces shaded or unshaded) denominator - (the total number of pieces) 5th Grade Math Notes Mixed Numbers and Improper Fractions When converting a mixed number into
More informationSquare Roots of Perfect Squares. How to change a decimal to a fraction (review)
Section 1.1 Square Roots of Perfect Squares How to change a decimal to a fraction (review) A) 0.6 The 6 is in the first decimal position called the tenths place. Therefore, B) 0.08 The 8 is in the second
More informationLesson #2: Exponential Functions and Their Inverses
Unit 7: Exponential and Logarithmic Functions Lesson #2: Exponential Functions and Their 1. Graph 2 by making a table. x f(x) -2.25-1.5 0 1 1 2 2 4 3 8 2. Graph the inverse of by making a table. x f(x).25-2.5-1
More informationFree Math print & Go Pages and centers. Created by: The Curriculum Corner.
Free Math print & Go Pages and centers Created by: The Curriculum Corner 1 x 3 9 x 9 4 x 5 6 x 7 2 x 1 3 x 7 8 x 4 5 x 9 4 x 6 8 x 8 7 x 2 9 x 3 1 x 5 4 x 4 8 x 3 4 x 8 8 x 10 5 x 5 1 x 8 4 x 3 6 x 6 8
More informationIntroduction to Fractions
Introduction to Fractions A fraction is a quantity defined by a numerator and a denominator. For example, in the fraction ½, the numerator is 1 and the denominator is 2. The denominator designates how
More informationLINEAR EQUATIONS IN TWO VARIABLES
LINEAR EQUATIONS IN TWO VARIABLES What You Should Learn Use slope to graph linear equations in two " variables. Find the slope of a line given two points on the line. Write linear equations in two variables.
More informationMAT 0002 Final Review A. Acosta. 1. Round to the nearest thousand. Select the correct answer: a b. 94,100 c. 95,000 d.
1. Round 94156 to the nearest thousand. 94000 94,100 95,000 d. 94,200 2. Round $67230 to the nearest $100. $68000 $67000 $67200 d. $67300 3. Subtract: 851 (476 61) 314 1,266 436 d. 446 PAGE 1 4. From the
More informationMAT 0002 Final Review A. Acosta
1. The page design for a magazine cover includes a blank strip at the top called a header, and a blank strip at the bottom called a footer. In the illustration below, how much page length is lost because
More informationUNIT 1: NATURAL NUMBERS.
The set of Natural Numbers: UNIT 1: NATURAL NUMBERS. The set of Natural Numbers ( they are also called whole numbers) is N={0,1,2,3,4,5...}. Natural have two purposes: Counting: There are three apples
More informationMath Review Packet. Grades. for th. Multiplication, Division, Decimals, Fractions, Metric & Customary Measurements, & Volume Math in the Middle
Math Review Packet for th 5 th 6 Grades Multiplication, Division, Decimals, Fractions, Metric & Customary Measurements, & Volume 206 Math in the Middle Multiplying Whole Numbers. Write the problem vertically
More informationCopyright 2015 Edmentum - All rights reserved.
Study Island Copyright 2015 Edmentum - All rights reserved. Generation Date: 05/19/2015 Generated By: Matthew Beyranevand Rounding Numbers 1. Round to the nearest hundred. 2,836 A. 2,900 B. 3,000 C. 2,840
More information5.4 Transformations and Composition of Functions
5.4 Transformations and Composition of Functions 1. Vertical Shifts: Suppose we are given y = f(x) and c > 0. (a) To graph y = f(x)+c, shift the graph of y = f(x) up by c. (b) To graph y = f(x) c, shift
More informationPROPERTIES OF FRACTIONS
MATH MILESTONE # B4 PROPERTIES OF FRACTIONS The word, milestone, means a point at which a significant change occurs. A Math Milestone refers to a significant point in the understanding of mathematics.
More informationMODULAR ARITHMETIC II: CONGRUENCES AND DIVISION
MODULAR ARITHMETIC II: CONGRUENCES AND DIVISION MATH CIRCLE (BEGINNERS) 02/05/2012 Modular arithmetic. Two whole numbers a and b are said to be congruent modulo n, often written a b (mod n), if they give
More information1 Write a Function in
www.ck12.org Chapter 1. Write a Function in Slope-Intercept Form CHAPTER 1 Write a Function in Slope-Intercept Form Here you ll learn how to write the slope-intercept form of linear functions and how to
More informationModule 5 Trigonometric Identities I
MAC 1114 Module 5 Trigonometric Identities I Learning Objectives Upon completing this module, you should be able to: 1. Recognize the fundamental identities: reciprocal identities, quotient identities,
More informationMATH STUDENT BOOK. 6th Grade Unit 6
MATH STUDENT BOOK 6th Grade Unit 6 Unit 6 Ratio, Proportion, and Percent MATH 606 Ratio, Proportion, and Percent INTRODUCTION 3 1. RATIOS 5 RATIOS 6 GEOMETRY: CIRCUMFERENCE 11 RATES 16 SELF TEST 1: RATIOS
More informationMath + 4 (Red) SEMESTER 1. { Pg. 1 } Unit 1: Whole Number Sense. Unit 2: Whole Number Operations. Unit 3: Applications of Operations
Math + 4 (Red) This research-based course focuses on computational fluency, conceptual understanding, and problem-solving. The engaging course features new graphics, learning tools, and games; adaptive
More informationLesson 5.4 Exercises, pages
Lesson 5.4 Eercises, pages 8 85 A 4. Evaluate each logarithm. a) log 4 6 b) log 00 000 4 log 0 0 5 5 c) log 6 6 d) log log 6 6 4 4 5. Write each eponential epression as a logarithmic epression. a) 6 64
More informationDeveloping Conceptual Understanding of Number. Set D: Number Theory
Developing Conceptual Understanding of Number Set D: Number Theory Carole Bilyk cbilyk@gov.mb.ca Wayne Watt wwatt@mts.net Vocabulary digit hundred s place whole numbers even Notes Number Theory 1 odd multiple
More informationAn ordered collection of counters in rows or columns, showing multiplication facts.
Addend A number which is added to another number. Addition When a set of numbers are added together. E.g. 5 + 3 or 6 + 2 + 4 The answer is called the sum or the total and is shown by the equals sign (=)
More informationPlace Value I. Number Name Standard & Expanded
Place Value I Number Name Standard & Expanded Objectives n Know how to write a number as its number name n Know how to write a number in standard form n Know how to write a number in expanded form Vocabulary
More informationLearn to solve multistep equations.
Learn to solve multistep equations. Levi has half as many comic books as Jamal has. If you add 6 to the number of comic books Jamal has and then divide by 7, you get the number of comic books Brooke has.
More informationMath 7 Notes Unit 02 Part A: Rational Numbers. Real Numbers
As we begin this unit it s a good idea to have an overview. When we look at the subsets of the real numbers it helps us organize the groups of numbers students have been exposed to and those that are soon
More informationLogarithms. In spherical trigonometry
Logarithms In spherical trigonometry there are many formulas that require multiplying two sines together, e.g., for a right spherical triangle sin b = sin B sin c In the 1590's it was known (as the method
More informationUnit 2: Exponents. 8 th Grade Math 8A - Mrs. Trinquero 8B - Dr. Taylor 8C - Mrs. Benefield
Unit 2: Exponents 8 th Grade Math 8A - Mrs. Trinquero 8B - Dr. Taylor 8C - Mrs. Benefield 1 8 th Grade Math Unit 2: Exponents Standards and Elements Targeted in the Unit: NS 1 Know that numbers that are
More informationCourse Syllabus - Online Prealgebra
Week 1 Week 2 Week 3 Week 4 Week 5 Week 6 1.1 Whole Numbers, Place Value Practice Problems for section 1.1 HW 1A 1.2 Adding Whole Numbers Practice Problems for section 1.2 HW 1B 1.3 Subtracting Whole Numbers
More informationCHM101 Lab Math Review and Significant Figures Grading Rubric
Name CHM101 Lab Math Review and Significant Figures Grading Rubric Criteria Points possible Points earned Part A (0.25 each) 3.5 Part B (0.25 each) 2.5 Part C (0.25 each) 1.5 Part D (Q5 0.25 each) 2 Part
More informationSection 2.1 Factors and Multiples
Section 2.1 Factors and Multiples When you want to prepare a salad, you select certain ingredients (lettuce, tomatoes, broccoli, celery, olives, etc.) to give the salad a specific taste. You can think
More informationRoots and Radicals Chapter Questions
Roots and Radicals Chapter Questions 1. What are the properties of a square? 2. What does taking the square root have to do with the area of a square? 3. Why is it helpful to memorize perfect squares?
More informationMinute Simplify: 12( ) = 3. Circle all of the following equal to : % Cross out the three-dimensional shape.
Minute 1 1. Simplify: 1( + 7 + 1) =. 7 = 10 10. Circle all of the following equal to : 0. 0% 5 100. 10 = 5 5. Cross out the three-dimensional shape. 6. Each side of the regular pentagon is 5 centimeters.
More informationMultiplying and Dividing Integers
Multiplying and Dividing Integers Some Notes on Notation You have been writing integers with raised signs to avoid confusion with the symbols for addition and subtraction. However, most computer software
More informationHomework 60: p.473: 17-45
8.4: Scientific Notation Homework 60: p.473: 17-45 Learning Objectives: Use Scientific Notation to represent extremely large and extremely small numbers Entry Task: Evaluate Each Expression (answer in
More informationFractions Presentation Part 1
New Jersey Center for Teaching and Learning Slide / Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and
More informationPlace Value (Multiply) March 21, Simplify each expression then write in standard numerical form. 400 thousands thousands = thousands =
Do Now Simplify each expression then write in standard numerical form. 5 tens + 3 tens = tens = 400 thousands + 600 thousands = thousands = Add When adding different units: Example 1: Simplify 4 thousands
More informationGrade 7 Math notes Unit 5 Operations with Fractions
Grade 7 Math notes Unit Operations with Fractions name: Using Models to Add Fractions We can use pattern blocks to model fractions. A hexagon is whole A trapezoid is of the whole. A parallelogram is of
More informationNumbers & Operations Chapter Problems
Numbers & Operations 8 th Grade Chapter Questions 1. What are the properties of a square? 2. What does taking the square root have to do with the area of a square? 3. Why is it helpful to memorize perfect
More information