LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II We just finished our review of linear functions. Linear functions are those that grow b equal differences for equal intervals. In this unit we will concentrate on eponential functions which grow b equal factors for equal intervals. To understand eponential functions, we first need to understand eponents. Eercise #: The following sequence shows powers of b repeatedl multipling b. Fill in the missing blanks. 9 This pattern can be duplicated for an base raised to an integer eponent. Because of this we can now define positive, negative, and zero eponents in terms of multipling the number repeatedl or dividing the number repeatedl. If n is an positive integer then: INTEGER EXPONENT DEFINITIONS... n-times n-times Eercise #: Given the eponential function f 0 evaluate each of the following without using our calculator. Show the calculations that lead to our final answer. f (b) f 0 (c) f (d) When increases b, b what factor does increase? Eplain our answer. emathinstruction, RED HOOK, NY 7, 0
Eercise #: (Propert #) For each of the following, write the product as a single eponential epression. Write and (b) as etended products first (if necessar). (b) (c) Generalize: 6 a b (d) Practice - Rewrite each of the following in simplest form: 0 (i) (ii) 6 (iii) 6 Eercise # (Propert #): Consider the epression Rewrite the quotient in simplest form b using b writing the numerator and denominator as etended products. a b. How can we simplif this quotient (division)? (b) Generalize for a b: a b (c) Practice - Rewrite each of the following in simplest form: (i) 8 (ii) 6 0 (iii) 6 Eercise # (Propert #): But what if the power of the numerator is less than that of the denominator? Rewrite the quotient as in Eercise #. (b) What results if ou appl Eponent Prop #? (c) Generalize: a (d) Rewrite each of the following without the use of negative eponents: (i) (ii) emathinstruction, RED HOOK, NY 7, 0
Eercise #6 (Propert #): What if the powers are the same? Rewrite the quotient based on the fundamental concept of dividing a quantit b itself. (b) What results if ou appl Eponent Prop #? (c) Generalize: 0 (d) Simplif each of the following: (i) 0 (ii) 0 Write each of the following with positive eponents onl. 7 (b) 0 (c) 6 The addition propert of eponents also etends to negative eponents as well. Write each of the following with positive eponents. (b) (c) a 7 a Eercise #7: (Propert #) For each of the following, write the eponential epression in the form. Write and (b) as etended products first (if necessar). (b) a (c) Generalize: ( ) b (c) Which of the following epressions is not equivalent to 0 () () 6 () () 6 0 0 0? The product propert of eponents also etends to negative eponents as well. (b) (c) emathinstruction, RED HOOK, NY 7, 0
Eercise #8 (Properties #6 and #7): Let's take a look at a. Rewrite using the definition of an (b) Generalize: eponent along with the associative and commutative properties of multiplication. a (c) Rewrite using the definition of an eponent along with the multiplication propert of fractions. (d) Generalize: a (e) Rewrite each of the following as equivalent epressions: (i) (ii) (iii) z emathinstruction, RED HOOK, NY 7, 0
LESSON #8 - INTEGER EXPONENTS COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Write each of the following eponential epressions without the use of eponents such as we did in lesson Eercise #. (b). Given the eponential function f evaluate each of the following without using our calculator. Show the calculations that lead to our final answer. f (b) f 0 (c) f emathinstruction, RED HOOK, NY 7, 0
6. Epress each of the following epressions in "epanded" form, i.e., do all of the multiplication and/or division possible and combine as man eponents as possible. (b) (c) 7 (d) 6 7 (e) 9 (f) 7 (g) 0 (h) 0 8 (i) 8 (j) 0 0 (k) (l) (m) (n) (o) (p). Which of the following is not equal to? Do not use our calculator to do this problem. () () 0. () () f. If 0 then f a () a () a () () a a emathinstruction, RED HOOK, NY 7, 0
7 8 6. Which of the following is equivalent to 6 () final answer. 6 9 () for all 0? Show the manipulations that lead to our () 6 9 () APPLICATIONS 7. It is helpful to be able to think about ver large numbers in terms of powers of 0. You should be familiar with man of these terms, but have ou thought about how man 0's are multipling each other? Here are some numbers to think about and eamples of things that would be counted in these quantities. Fill in the proper power of 0. The first has been done for ou. NUMBER POWER OF 0 EXAMPLE million billion 6,000,000 0 0 0 6,000 million 0 0 6 trillion million 0 quadrillion 000 trillion quintillion billion The distance between New York Cit and Boston is approimatel million feet. There are approimatel billion seconds in a centur. There are 6 trillion miles in a light ear, i.e. the distance light can travel in a ear. There are approimatel quadrillion ants populating the earth at an time. There are approimatel 8 quintillion grains of sand on all of the Earth's beaches. REASONING 8. We've etended the two fundamental eponent properties to negative as well as positive integers. What would happen if we etended the Product Eponent Propert to a fractional eponent like? Let's pla around with that idea. Use the Product Propert of Eponents to justif that 9 9. (b) What other number can ou square that results in 9? Hmm... emathinstruction, RED HOOK, NY 7, 0
8 LESSON #9 - RATIONAL EXPONENTS COMMON CORE ALGEBRA II When ou first learned about eponents, the were alwas positive integers, and just represented repeated multiplication. And then we had to go and introduce negative eponents, which reall just represent repeated division. Toda we will introduce rational (or fractional) eponents and etend our eponential knowledge that much further. Eercise #: Recall the Product Propert of Eponents and use it to rewrite each of the following as a simplified eponential epression. There is no need to find a final numerical value. (b) 7 (c) 0 (d) We will now use the Product Propert to etend our understanding of eponents to include unit fraction eponents (those of the form n where n is a positive integer). Eercise #: Consider the epression 6. Appl the Product Propert to simplif 6. What other number squared ields 6? (b) You can now sa that 6 is equivalent to what more familiar quantit? This is remarkable! An eponent of is equivalent to a square root of a number!!! Eercise #: Test the equivalence of the eponent to the square root b using our calculator to evaluate each of the following. Be careful in how ou enter each epression. (b) 8 (c) 00 We can etend this now to all levels of roots, that is square roots, cubic roots, fourth roots, etcetera. UNIT FRACTION EXPONENTS For n given as a positive integer: emathinstruction, RED HOOK, NY 7, 0
9 So far ou have onl used the square root or nd root. There are also cube or rd roots, th roots, etc. The following is the notation for roots where n is the root: n Cube root ( rd root) notation: th root notation:. n is called the inde of the root. Eercise #: Just as there are perfect squares, there are also perfect cubes, powers of four, etc. These numbers are important to know so that ou can begin working with higher roots. a) List the perfect cubes up to 6 (don t forget the negatives): Odd powers can have negative values, but even powers are alwas positive because the negatives cancel out. b) List the perfect th powers up to c) List the perfect th powers up to : (don t forget the negatives): ( )( )( ) 8 ( )( )( )( ) 6 d) List the perfect 6 th powers up to 6 : These numbers will need to become familiar to ou just as the perfect squares alread are. Eercise #: Simplif each of the following roots. a) 7 b) 8 c) 6 d) 6 e) f) 6 emathinstruction, RED HOOK, NY 7, 0
Eercise #6: Rewrite each of the following using roots instead of fractional eponents. Then, if necessar, evaluate using our calculator to guess and check to find the roots (don't use the generic root function). Check with our calculator. 0 (b) 6 (c) 9 (d) We can now combine traditional integer powers with unit fractions in order interpret an eponent that is a rational number, i.e. the ratio of two integers. The net eercise will illustrate the thinking. Remember, we want our eponent properties to be consistent with the structure of the epression. Eercise #7: Let's think about the epression. Fill in the missing blank and then evaluate this epression: (b) Fill in the missing blank and then evaluate this epression: (c) Verif both and (b) using our calculator. (d) Evaluate 7 without our calculator. Show our thinking. Verif with our calculator. RATIONAL EXPONENT CONNECTION TO ROOTS For the rational number we define to be: or. Eercise #8: Evaluate each of the following eponential epressions involving rational eponents without the use of our calculator. Show our work. Then, check our final answers with the calculator. 6 (b) (c) 8 emathinstruction, RED HOOK, NY 7, 0
LESSON #9 - RATIONAL EXPONENTS COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Rewrite the following as equivalent roots and then evaluate as man as possible without our calculator. 6 (b) 7 (c) (d) 00 (e) 6 (f) 9 (g) 8 (h). Evaluate each of the following b considering the root and power indicated b the eponent. Do as man as possible without our calculator. 8 (b) (c) 6 (d) 8 (e) (f) 7 8 (g) 6 (h). Given the function f, which of the following represents its -intercept? () 0 () () 0 () 0 emathinstruction, RED HOOK, NY 7, 0
. Which of the following is equivalent to () () () ()?. Written without fractional or negative eponents, () () () () is equal to 6. Which of the following is not equivalent to () 096 () 6 6? () 8 () 6 REASONING 7. Marlene claims that the square root of a cube root is a sith root? Is she correct? To start, tr rewriting the epression below in terms of fractional eponents. Then appl the Power Propert of Eponents. a 8. We should know that 8. To see how this is equivalent to do this, we can rewrite the equation as: How can we now use this equation to see that n 8? 8 we can solve the equation 8 n. To emathinstruction, RED HOOK, NY 7, 0
LESSON #0 - FRACTIONAL EXPONENTS REVISITED COMMON CORE ALGEBRA II Recall that in Unit # we introduced the concept that roots (square roots, cube roots, etcetera) could be represented b rational or fractional eponents. For n given as a positive integer: Eercise #: Rewrite each epression in the form UNIT FRACTION EXPONENTS b a where a and b are both rational numbers. (b) (c) 7 (d) 0 Recall we can also combine integer powers with roots with the following: RATIONAL EXPONENT CONNECTION TO ROOTS For the rational number, is equivalent to: or. Power on Top. Roots below the Ground. Eercise #: Rewrite each of the following power/root combinations as a rational eponent in simplest form. 7 (b) 6 (c) 6 (d) 0 Eercise #: If 0, then which of the following represents the value of f f without the use of a calculator. Show the steps in our calculation. () 6 () () 8 () 7 Eercise #: Which of the following is not equivalent to 7/?? Find the value () 7 () 7 () 7 () 7 emathinstruction, RED HOOK, NY 7, 0
Eercise #: Simplif each of the following roots. a) 0 b) 6 8 c) 7 d) 6 0 e) a f) 0 0 80a b g) 8 6 h) i) 8 8 j) k) 0 0 0 l) 6 6 6 Fractional eponents pla b the same rules (properties) as all other eponents. It is, in fact, these properties that can justif man standard manipulations with square roots (and others). For eample, simplifing roots. Eercise #6: We onl consider a square root "simplified" when all of its perfect square factors have had their square roots evaluated. Fill in the eponent propert below: ab n (b) Rewrite 8 in factored form, with one factor being the largest perfect square divisor. Also, write the square root in eponent form. (c) Simplif 8 using (b) and the propert from. (d) Generalize: ab n ab emathinstruction, RED HOOK, NY 7, 0
Eercise #7: Simplif each of the following square roots. Show the manipulations that lead to our answers. 8 (b) 00 (c) 9 7 We can etend the simplifing process to include cube roots and higher-order roots b simpl etending our thinking. Eercise #8: Simplif each of the following higher order roots. 6 (b) 08 (c) 0 (d) 8 8 (e) 6 (f) 6 8 (g) 0 8 (h) 6 emathinstruction, RED HOOK, NY 7, 0
6 LESSON #0 - FRACTIONAL EXPONENTS REVISITED COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Which of the following is equivalent to? () () () (). If the epression was placed in a form, then which of the following would be the value of a? () () () (). Which of the following is not equivalent to 9? () () 9 () 9 (). The radical epression 0 can be rewritten equivalentl as () () () () 0. If the function was placed in the form b a then which of the following is the value of ab? () 6 () 6 () () emathinstruction, RED HOOK, NY 7, 0
7 6. Rewrite each of the following epressions without roots b using fractional eponents. (b) (c) 7 (d) (e) (f) (g) (h) 9 7. Rewrite each of the following without the use of fractional or negative eponents b using radicals. 6 (b) 0 (c) (d) (e) (f) 7 (g) 9 (h) 8. Simplif each of the following square roots that contain variables in the radicand. 9 8 (b) 6 7 (c) 8 7 (d) 98 8 9. Epress each of the following roots in simplest radical form. 6 8 (b) 0 08 (c) 6 (d) 7 7 0. Mikala was tring to rewrite the epression incorrectl rewrote it as. Eplain Mikala's error. in an equivalent form that is more convenient to use. She emathinstruction, RED HOOK, NY 7, 0
8 LESSON # - MORE EXPONENT PRACTICE COMMON CORE ALGEBRA II For further stud in mathematics, especiall Calculus, it is important to be able to manipulate epressions involving eponents, whether those eponents are positive, negative, or fractional. The basic laws of eponents, which ou should have learned in Algebra and have used previousl in this course, are shown to the right. The appl regardless of the nature of the eponent (i.e. positive, negative, or fractional). EXPONENT LAWS... 6.. 7.. (For integers m and n) Although these problems can be challenging, the ke will be to carefull appl these eponent laws in a sstematic manner. Eercise #: Simplif each of the following epressions. Leave no negative eponents in our answers. (b) 7 (c) (d) 6 In the last eercise, all of the powers were integers. In the net eercise, we introduce fractional powers. Remember, though, that the will still follow the eponent rules above. If needed, use our calculator to help add and subtract the powers. Eercise #: Simplif each of the following epressions. Write each without the use of negative eponents. (b) (c) 6 8 emathinstruction, RED HOOK, NY 7, 0
We must not forget from our last lesson that fractional eponents have an equivalent interpretation as roots. We should be able to move from one representation to another. Eercise #: Rewrite each epression below using radical epressions in simplest form. 9 (b) (c) (d) (e) 7 8 (f) 6 Eercise #: Which of the following is equivalent to 8 7? () () 7 8 () 7 () 7 7 8 Eercise #: The epression () () () () is the same as emathinstruction, RED HOOK, NY 7, 0
0 LESSON # MORE EXPONENT PRACTICE COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Rewrite each of the following epressions in simplest form and without negative eponents. 7 (b) (c) (d) 0 8. Which of the following represents the value of () 9 () 6 a b when a and b? () 8 (). Simplif each epression below so that it contains no negative eponents. Do not write the epressions using radicals. 7 (b) (c). Which of the following represents the epression () () () () 6 written in simplest form? emathinstruction, RED HOOK, NY 7, 0
. Rewrite each of the following epressions using radicals. Epress our answers in simplest form. (b) (c) (d) (e) (f) 6. Which of the following is equivalent to 0? () () () () 7. When written in terms of a fractional eponent the epression () 7 () is () () 8. Epressed as a radical epression, the fraction () 6 () 6 () 6 () 6 is emathinstruction, RED HOOK, NY 7, 0
LESSON # - THE METHOD OF COMMON BASES COMMON CORE ALGEBRA II There are ver few algebraic techniques that do not involve technolog to solve equations that contain eponential epressions. In this lesson we will look at one of the few, known as The Method of Common Bases. Eercise #: Solve each of the following simple eponential equations b writing each side of the equation using a common base. Most common bases are,,,, and. 6 (b) 7 (c) (d) 6 In each of these cases, even the last, more challenging one, we could manipulate the right-hand side of the equation so that it shared a common base with the left-hand side of the equation. We can eploit this fact b manipulating both sides so that the have a common base. First, though, we need to review an eponent law. Eercise #: Simplif each of the following eponential epressions. (b) (c) 7 (d) Eercise #: Solve each of the following equations b finding a common base for each side. 8 (b) 9 7 (c) Eercise #: Which of the following represents the solution set to the equation () () 0, () () 6? emathinstruction, RED HOOK, NY 7, 0
This technique can be used in an situation where all bases involved can be written with a common base. In a practical sense, this is rather rare. Yet, these tpes of algebraic manipulations help us see the structure in eponential epressions. Tr to tackle the net, more challenging, problem. Eercise #: Two eponential curves, and are shown below. The intersect at point A. A rectangle has one verte at the origin and the other at A as shown. We want to find its area. Fundamentall, what do we need to know about a rectangle to find its area? (b) How would knowing the coordinates of point A help us find the area? A (c) Find the area of the rectangle algebraicall using the Method of Common Bases. Show our work carefull. Eercise #6: At what coordinate will the graph of work that leads to our choice. a intersect the graph of? Show the () () a () a () a a emathinstruction, RED HOOK, NY 7, 0
LESSON # - THE METHOD OF COMMON BASES COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Solve each of the following eponential equations using the Method of Common Bases. Each equation will result in a linear equation with one solution. Check our answers. 9 (b) 7 6 (c) (d) (e) 6 8 (f) 96. Algebraicall determine the intersection point of the two eponential functions shown below. Recall that most sstems of equations are solved b substitution. 8 and 9. Algebraicall determine the zeroes of the eponential function known as a zero is because the output is zero. f. Recall that the reason it is emathinstruction, RED HOOK, NY 7, 0
APPLICATIONS. One hundred must be raised to what power in order to be equal to a million cubed? Solve this problem using the Method of Common Bases. Show the algebra ou do to find our solution.. The eponential function intersection point is a, 0 is shown graphed along with the horizontal line. Their. Use the Method of Common Bases to find the value of a. Show our work. a, 0 REASONING 6 The Method of Common Bases works because eponential functions are one-to-one, i.e. if the outputs are the same, then the inputs must also be the same. This is what allows us to sa that if, then must be equal to. But it doesn't alwas work out so easil. If, can we sa that must be? Could it be anthing else? Wh does this not work out as easil as the eponential case? emathinstruction, RED HOOK, NY 7, 0
6 LESSON # - EXPONENTIAL FUNCTION BASICS COMMON CORE ALGEBRA II You studied eponential functions etensivel in Common Core Algebra I. Toda's lesson will review man of the basic components of their graphs and behavior. Eponential functions, those whose eponents are variable, are etremel important in mathematics, science, and engineering. BASIC EXPONENTIAL FUNCTIONS where Eercise #: Consider the function the graph on the grid provided.. Fill in the table below without using our calculator and then sketch 0 Eercise #: Now consider the function and sketch the graph on the aes provided.. Using our calculator to help ou, fill out the table below Notice: The eponential function is increasing when base>. 0 Notice: The eponential function is increasing when base is between 0 and. emathinstruction, RED HOOK, NY 7, 0
7 Eercise #: Based on the graphs and behavior ou saw in Eercises # and #, state the domain and range for an eponential function of the form b. Domain (input set): Range (output set): There are man numbers in mathematics that are more important than others because the find so man uses in either mathematics or science. Good eamples of important numbers are 0,, i, and. In this lesson ou will be introduced to an important number given the letter e for its inventor Leonhard Euler (707-78). This number plas a crucial role in Calculus and more generall in modeling eponential phenomena. We will be working more with e in the net unit. Toda we will just be graphing the function, e. THE NUMBER e. Like, e is irrational.. e. Used in Eponential Modeling Eerise #: Now consider the function sketch the graph on the aes provided. 0 e e. Using our calculator to help ou, fill out the table below and Eercise #: Are all eponential functions one-to-one? How can ou tell? What does this tell ou about their inverses? emathinstruction, RED HOOK, NY 7, 0
8 Eercise #6: Now consider the function 7. Determine the -intercept of this function algebraicall. Justif our answer. (b) Does the eponential function increase or decrease? Eplain our choice. (c) Create a rough sketch of this function, labeling its - intercept.. Eercise #7: Consider the function How does this function s graph compare to that of? What does adding do to a function's graph? (b) Determine this graph s -intercept algebraicall. Justif our answer. (c) Create a rough sketch of this function, labeling its - intercept. 8. The graph below shows two eponential functions, with real number constants a, b, c, and d. Given the graphs, onl one pair of the constants shown below could be equal in value. Determine which pair could be equal and eplain our reasoning. c and d a and b a and c a c b d emathinstruction, RED HOOK, NY 7, 0
9 LESSON # - EXPONENTIAL FUNCTION BASICS COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Which of the following represents an eponential function? () 7 () () 7 () f then f. If 69 a calculator.) 7 7? (Remember what we just learned about fractional eponents and do without () 7 () 7 () 8 (). If h and g 7 then h g () 8 () 8 () () 7. Which of the following equations could describe the graph shown below? () () () (). Which of the following equations represents the graph shown? () () () () 0 - - - emathinstruction, RED HOOK, NY 7, 0
0 6. Sketch graphs of the equations shown below on the aes given. Label the -intercepts of each graph. 8 (b) APPLICATION 7. The Fahrenheit temperature of a cup of coffee, F, starts at a temperature of 8 F. It cools down according to the eponential function Fm m 0 How do ou interpret the statement that F 60 86? 7, where m is the number minutes it has been cooling. (b) Determine the temperature of the coffee after one da using our calculator. What do ou think this temperature represents about the phsical situation? REASONING 8. The graph below shows two eponential functions, with real number constants a, b, c, and d. Given the graphs, onl one pair of the constants shown below could be equal in value. Determine which pair could be equal and eplain our reasoning. b and d a and b a and c a b c d 9. Eplain wh the equation below can have no real solutions. If ou need to, graph both sides of the equation using our calculator to visualize the reason. emathinstruction, RED HOOK, NY 7, 0
LESSON # - INTRODUCTION TO LOGARITHMS COMMON CORE ALGEBRA II Eponential functions are of such importance to mathematics that their inverses, functions that reverse their action, are important themselves. These functions, known as logarithms, will be introduced in this lesson. Eercise #: The function is shown graphed on the aes below along with its table of values. 0 8 8 Is this function one-to-one? Eplain our answer. (b) Based on our answer from part, what must be true about the inverse of this function? (c) Create a table of values below for the inverse of and plot this graph on the aes given. Notice that, as alwas, the graphs of a function and its inverse are smmetric across. (d) What would be the first step to find an equation for this inverse algebraicall? Write this step down and then stop. Defining Logarithmic Functions The function eample, log is the inverse of. log b is the name we give the inverse of b. For An eponential equation is set up in the form, eponent base An equivalent logarithmic equation is set up in the form, log base number log b n Therefore, e or Bne. b is the same as log = b number. We can nickname this emathinstruction, RED HOOK, NY 7, 0 e b n or Ben. eponent We can nickname this Based on this, we see that a logarithm gives as its output (-value) the eponent we must raise b to in order to produce its input (-value).
Eercise #: Evaluate the following logarithms. If needed, write an equivalent eponential equation. Do as man as possible without the use of our calculator. log 8 (b) log 6 (c) log 6 (d) log000,000 (e) log6 6 (f) log 6 (g) log (h) log 9 It is criticall important to understand that logarithms give eponents as their outputs. We will be working for multiple lessons on logarithms and a basic understanding of their inputs and outputs is critical. Eercise #: If the function would represent its -intercept? () () 8 () () 9 log 8 9 was graphed in the coordinate plane, which of the following Eercise #: Between which two consecutive integers must log 0 lie? () and () and () and () and Calculator Use and Logarithms Most calculators onl have two logarithms that the can evaluate directl. One of them, log0, is so common that it is actuall called the common log and tpicall is written without the base 0. log log 0 (The Common Log) Eercise #: Evaluate each of the following using our calculator. log00 (b) log 000 (c) log 0 emathinstruction, RED HOOK, NY 7, 0
LESSON # - INTRODUCTION TO LOGARITHMS COMMON CORE ALGEBRA II HOMEWORK FLUENCY. Which of the following is equivalent to log7? () 7 () 7 () 7 () 7. If the graph of 6 is reflected across the line then the resulting curve has an equation of () 6 () log6 () log6 () 6. The value of log 67 is closest to which of the following? Hint guess and check the answers. ().67 ().8 ().98 ().8. Which of the following represents the -intercept of the function () 8 () () () log 000 8?. Determine the value for each of the following logarithms. (Eas) log (b) log7 9 (c) log66 (d) log 0 6. Determine the value for each of the following logarithms. (Medium) log 6 (b) log (c) log (d) log7 emathinstruction, RED HOOK, NY 7, 0
7. Determine the value for each of the following logarithms. Each of these will have non-integer, fractional answers. (Difficult) log (b) log 8 (c) log (d) log 8. Between what two consecutive integers must the value of log 7 lie? Justif our answer. 9. Between what two consecutive integers must the value of log 00 lie? Justif our answer. APPLICATIONS 0. In chemistr, the ph of a solution is defined b the equation ph log H where H represents the concentration of hdrogen ions in the solution. An solution with a ph less than 7 is considered acidic and an solution with a ph greater than 7 is considered basic. Fill in the table below. Round our ph s to the nearest tenth of a unit. (Note: this is the common log). Substance Concentration of Hdrogen ph Basic or Acidic? 7 Milk.6 0 Coffee. 0 Bleach. 0 Lemmon Juice 7.9 0 6 Rain.6 0 REASONING. Can the value of log ou about the domain of log b? be found? What about the value of log 0? Wh or wh not? What does this tell emathinstruction, RED HOOK, NY 7, 0
LESSON # - GRAPHS OF LOGARITHMS COMMON CORE ALGEBRA II The vast majorit of logarithms that are used in the real world have bases greater than one; the ph scale that we saw on the last homework assignment is a good eample. In this lesson we will further eplore graphs of these logarithms, including their construction, transformations, and domains and ranges. Eercise #: Consider the logarithmic function log and its inverse. Construct a table of values for and then use this to construct a table of values for the function log. 0 log (b) Graph and log on the grid given. Show the algebraic process of finding the inverse of to get log. (c) Determine the and intercepts of each function. log -intercept -intercept (d) State the domain and range of and log. Domain: Range: Domain: Range: log emathinstruction, RED HOOK, NY 7, 0
6 An asmptote is a line that a graph approaches but never crosses. (e) What is the asmptote of the function? (f) What is the asmptote of the function log? Eercise #: Using our calculator, sketch the graph of log0 on the aes below. Label the -intercept. State the domain and range of log0. Domain: Range: 0 Eercise #: What are the and intercepts of the function, log 9? The fact that finding the logarithm of a non-positive number (negative or zero) is not possible in the real number sstem allows us to find the domains of a variet of logarithmic functions. Eercise #: Determine the domain of the function log. State our answer in set-builder notation. Eercise #: Which of the following values of is not in the domain of f log () () 0 () () -? emathinstruction, RED HOOK, NY 7, 0
7 Eercise #6: Given the function, e, Algebraicall, find the function s inverse. Just as e is a ver important number in eponential modeling, log e also arises in real world problems ver often. It is called the natural logarithm and is usuall epressed as ln. In other words, log ln. Therefore, e and ln are inverses of each other. e (b) Construct a table of values for e and then use this to construct a table of values for the function ln. Round values to the nearest tenth. 0 e ln Eercise #7: Without the use of our calculator, determine the values of each of the following. ln e (b) ln (c) ln e (d) ln e emathinstruction, RED HOOK, NY 7, 0
8 LESSON # - GRAPHS OF LOGARITHMS COMMON CORE ALGEBRA II HOMEWORK FLUENCY. The domain of log in the real numbers is () 0 () () (). Which of the following equations describes the graph shown below? (Hint: Plug in (,) to see which equation works). () log () log () log () log. Which of the following represents the -intercept of the function () 8 () () () log?. Which of the following values of is not in the domain of f log 0 () () () 0 (). Which of the following is true about the function () It has an -intercept of and a -intercept of. () It has -intercept of and a -intercept of. () It has an -intercept of 6 and a -intercept of. () It has an -intercept of 6 and a -intercept of. log 6?? emathinstruction, RED HOOK, NY 7, 0
9 6. Which of the following is closest to the -intercept of the function whose equation is () 0 () 7 () 8 () 0e? 7. On the grid below, the solid curve represents e describe the dashed curve? Eplain our choice. () (). Which of the following eponential functions could () e () 8. Determine the domains of each of the following logarithmic functions. State our answers using an accepted notation. Be sure to show the inequalit that ou are solving to find the domain and the work ou use to solve the inequalit. log (b) log 6 9. Graph the logarithmic function log on the graph paper given. For a method, see Eercise # on page. emathinstruction, RED HOOK, NY 7, 0
0 0. Without the use of our calculator, determine the values of each of the following. ln e (b) ln e (c) ln e (d) 0 ln e e 0 REASONING. Logarithmic functions whose bases are larger than tend to increase ver slowl as increases. Let's investigate this for f log. Find the value of f, f, f, and f 8 without our calculator. (b) For what value of will? For what value of will log 0 log 0? emathinstruction, RED HOOK, NY 7, 0