Math / 11th Grade /
Unit 5: Exponential Modeling and Logarithms
Exponential Modeling and Logarithms
Students revisit exponential functions, including geometric sequences and series, and learn to manipulate logarithmic expressions and equations to solve problems involving exponential modeling.
Students have previously seen exponential functions in Algebra I. This unit builds off of that knowledge, revisiting exponential functions and including geometric sequences and series and continuous compounding situations. In the second part of the unit, students learn that the logarithm is the inverse of the exponent and to manipulate logarithmic expressions and equations. Finally, students apply their knowledge of logarithms to solve problems involving exponential modeling.
This unit is an excellent opportunity for students to practice mathematical modeling using exponential functions as models for situations in the world. Students will also look for and make use of structure as they manipulate logarithms and connect their knowledge of exponents to logarithms.
While this unit culminates the study of exponents and logarithms in the Common Core State Standards, it leads to essential topics in calculus. Students preparing to take a calculus course should emphasize algebraic manipulation of exponential and logarithmic expressions to rewrite them in a variety of ways, including using properties of logarithms and analyzing functions to connect their graphs to equations and contextual situations.
The central mathematical concepts that students will come to understand in this unit
Write expressions or equations or change the form of an equation to model contexts and connect components of those expressions and equations to graphs and contexts.
Explain that logarithmic functions are the inverse of exponential functions and have parallel properties to exponents used to rewrite expressions and solve equations. Know that the properties of logarithms can be used to change the form of equations to reveal solution paths.
Use knowledge of exponents and logarithms to solve exponential modeling problems where a piece of information is missing.
Vocabulary
Terms and notation that students learn or use in the unit
Geometric sequence
Geometric series
Compounding/Continuous compounding
Percentage Rate
e (Euler's number)
Base
Rate
Argument
Principal
Finite geometric series
Sum of geometric series
Summation notation $${\sum}$$
Logarithm
Natural log
Common log
Change of base
Product property of logarithms (Logarithm law)
Quotient property (Logarithm law)
Power property (Logarithm law)
Exponentiate
Exponential growth
Exponential decay
Materials
The materials, representations, and tools teachers and students will need for this unit
Equations, tables, graphs, and contextual situations
A calculator or other technology to graph and solve problems using exponential modeling and logarithms
Lesson Map
Topic A: Modeling with and Interpreting Exponential Functions
F.BF.A.1— Write a function that describes a relationship between two quantitiesModeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Building Functions
F.BF.A.1.A— Determine an explicit expression, a recursive process, or steps for calculation from a context.
Building Functions
F.BF.B.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.
Building Functions
F.BF.B.4— Find inverse functions.
Building Functions
F.BF.B.4.B— Verify by composition that one function is the inverse of another.
Building Functions
F.BF.B.4.C— Read values of an inverse function from a graph or a table, given that the function has an inverse.
Building Functions
F.BF.B.5— Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
Interpreting Functions
Interpreting Functions
F.IF.A.3— Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1.
Interpreting Functions
F.IF.B.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.
Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Interpreting Functions
F.IF.B.5— Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Interpreting Functions
F.IF.C.7— Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Interpreting Functions
F.IF.C.7.E— Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Interpreting Functions
F.IF.C.8— Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Interpreting Functions
F.IF.C.8.B— Use the properties of exponents to interpret expressions for exponential functions.For example, identify percent rate of change in functions such as y = (1.02)<sup>t</sup>, y = (0.97)<sup>t</sup>, y = (1.01 <sup>12t</sup>, y = (1.2)<sup>t/10</sup>, and classify them as representing exponential growth or decay.
Linear, Quadratic, and Exponential Models
Linear, Quadratic, and Exponential Models
F.LE.A.2— Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Linear, Quadratic, and Exponential Models
F.LE.A.4— For exponential models, express as a logarithm the solution to ab<sup>ct</sup> = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
Linear, Quadratic, and Exponential Models
F.LE.B.5— Interpret the parameters in a linear or exponential function in terms of a context.
Seeing Structure in Expressions
Seeing Structure in Expressions
A.SSE.A.1— Interpret expressions that represent a quantity in terms of its contextModeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Seeing Structure in Expressions
A.SSE.A.1.B— Interpret complicated expressions by viewing one or more of their parts as a single entity.For example, interpret P(1+r)<sup>n</sup> as the product of P and a factor not depending on P.
Seeing Structure in Expressions
A.SSE.B.3— Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Seeing Structure in Expressions
A.SSE.B.3.C— Use the properties of exponents to transform expressions for exponential functions.For example the expression 1.15<sup>t</sup> can be rewritten as (1.151/12)<sup>12t</sup> 1.012<sup>12t</sup> to reveal the approximate equivalent monhly interest rate if the annual rate is 15%.
Seeing Structure in Expressions
A.SSE.B.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.
Modeling is best interpreted not as a collection of isolated topics but in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★). The star symbol sometimes appears on the heading for a group of standards; in that case, it should be understood to apply to all standards in that group.
Foundational Standards
Standards covered in previous units or grades that are important background for the current unit
High School — Number and Quantity
High School — Number and Quantity
N.RN.A.1— Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.For example, we define 5<sup>1/3</sup> to be the cube root of 5 because we want (5<sup>1/3</sup>)³ = 5(<sup>1/3</sup>)³ to hold, so (5<sup>1/3</sup>)³ must equal 5.
High School — Number and Quantity
N.RN.A.2— Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Linear, Quadratic, and Exponential Models
Linear, Quadratic, and Exponential Models
F.LE.A.1— Distinguish between situations that can be modeled with linear functions and with exponential functions.
Linear, Quadratic, and Exponential Models
F.LE.A.1.A— Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.
Linear, Quadratic, and Exponential Models
F.LE.A.1.B— Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
Linear, Quadratic, and Exponential Models
F.LE.A.1.C— Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.
Reasoning with Equations and Inequalities
Reasoning with Equations and Inequalities
A.REI.A.1— Explain each step in solving a simple equation as following from the equality of numbers 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.
Seeing Structure in Expressions
Seeing Structure in Expressions
A.SSE.A.2— Use the structure of an expression to identify ways to rewrite it.For example, see x<sup>4</sup> — y<sup>4</sup> as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).
Standards for Mathematical Practice
CCSS.MATH.PRACTICE.MP1— Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2— Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3— Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4— Model with mathematics.
CCSS.MATH.PRACTICE.MP5— Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6— Attend to precision.
CCSS.MATH.PRACTICE.MP7— Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8— Look for and express regularity in repeated reasoning.
