Math 3 Unit 4a: Logarithms and Exponents Representations



Approximate Time Frame: 3 – 4 WeeksConnections to Previous Learning: In previous years, students have studied exponential functions, but have only been able to solve equations with an unknown in the exponent that can be solved by inspection or the use of exponent rules. Logarithms open the door to students being able to solve more such equations, not restricted to trivial cases. This unit builds on students’ understanding of linear, quadratic, exponential, polynomial, and rational functions to include logarithmic functions as well.Focus of this Unit: In this unit, students will learn the features of logarithmic functions, focusing on graphs and the relationship between logarithms and exponentials. A key understanding is that a logarithm is itself an exponent, and therefore it can be the solution to an exponential equation. Connections to Subsequent Learning: In the next unit, students will compare logarithms and exponentials with the other types of functions they have studied and apply them in solving problems.589851510795From the High School, Algebra Progression document p.5:Much of the ability to see and use structure in transforming expressions comes from learning to recognize certain fundamental techniques. One such technique is recognizing internal cancellations, as in the expansion a-ba+b=a2-b2An impressive example of this is x-1xn-1+xn-2+…+x+1=xn-1in which all the terms cancel except the end terms. This identity is the foundation for the formula for the sum of a finite geometric series.A-SSE.4From the Grade 8, High School, Functions Progression document p.15: Students note the correspondence between rise and run on agraph and differences of inputs and outputs in a symbolic form of the proof (MP1). A symbolic proof has the advantage that the analogous proof showing that exponential functions grow by equal factors over equal intervals begins in an analogous way.InputOutput05833The process of going from linear or exponential functions to tables can go in the opposite direction. Given sufficient information, e.g., a table of values together with information about the type of relationship represented, F-LE.4, students construct the appropriate function. For example, students might be given the information that the table below shows inputs and outputs of an exponential function, and asked to write an expression for the function. Desired OutcomesStandard(s):Create equations that describe numbers or relationships.A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. Understand solving equations as a process of reasoning and explain the reasoning.A.REI.1 Explain each step in solving a simple equation as following from the equality of number asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.Interpret functions that arise in application in terms of the context.F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Analyze functions using different representations.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.Build new functions from existing functions. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.Write expressions in equivalent forms to solve problems. A.SSE.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1) and use the formula to solve problems. For example, calculate mortgage payments. Construct and compare linear, quadratic, and exponential models and solve problems.F.LE.4 For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. WIDA Standard: (English Language Learners)English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics.English language learners benefit from:Explicit vocabulary instruction with regard to the types and components of function representations.Guided conversations regarding the connections between graphic, algebraic, tabular and verbal descriptions of functions.Understandings: Students will understand …The logarithm of a number is the exponent that another value (the base) must be raised to produce the given number.That is another way of expressing and that this logarithmic expression can be used to determine the solution of an equation where the unknown is in the exponent.Exponential functions and equations can be rewritten as logarithmic functions and equations, and vice versa.The graphs of logarithmic functions have key features, including domain, intercepts, and end behavior.Transformations on the function modify the key features of the function.Convenient values for representing a logarithmic function on a table are powers of the base of the logarithm.Essential Questions:What is a logarithm?How are logarithms and exponentials related?How can exponential equations be solved for an unknown in the exponent?What are the key features of the graph of a logarithmic function?How can a logarithmic function be represented numerically or in a table?Mathematical Practices: (Practices to be explicitly emphasized are indicated with an *.)1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. *5. Use appropriate tools strategically. Students will develop this mathematical practice as they determine which logarithmic expressions can be evaluated with and without the use of technology.6. Attend to precision. *7. Look for and make use of structure. Students will develop this mathematical practice as they further investigate the use of transformations on functions (logarithmic) and relate a function to its graph and a table of values. 8. Look for and express regularity in repeated reasoning. Prerequisite Skills/Concepts: Students should already be able to:Write, graph, and interpret exponential functions.Solve equations with unknowns in the exponent by inspection or use of exponent rules.Identify the effects of the transformations f(x) + k, k f(x), f(kx), and f(x + k).Advanced Skills/Concepts:Some students may be ready to:Use logarithms to give exact solutions to exponential equations in any base.Understand logarithms and exponentials as inverse functions and use this to explain the relationships between domain, range, asymptotes, and average rate of change.Use properties of logarithms to rewrite expressions.F.BF.4 Find Inverse functions.b. (+) Verify by composition that one function is the inverse ofanother.c. (+) Read values of an inverse function from a graph or a table,given that the function has an inverse.d. (+) Produce an invertible function from a non-invertible functionby restricting the domain.F.BF.5. (+) Understand the inverse relationship between exponents andlogarithms and use this relationship to solve problems involvinglogarithms and exponents.Knowledge: Students will know…The definition of a logarithm.The graph of f(x) = log (x) has a domain of x>0, a vertical asymptote at x=0, and an x-intercept at x = 1.Change of base formula.Skills: Students will be able to …Use logarithms to solve exponential equations in base 2, 10, or e.Evaluate logarithms based on the definition for simple cases.Evaluate logarithms using the change of base formula with technology.Graph exponential functions, identifying intercepts and end behavior.Graph logarithmic functions, identifying intercepts and end behavior.Construct a viable argument to justify a solution method.Calculate and interpret the average rate of change of a function.Translate back and forth between logarithmic and exponential pare properties of two functions each represented in a different way.Derive the formula for the sum of a finite geometric series.Academic Vocabulary:Critical Terms:LogarithmBaseAsymptoteNatural logarithm (ln or loge)Supplemental Terms:Inverse functionLimit ................
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