Big Idea:

Big Idea 4

A logarithm is the inverse of an exponential function.

1 week

Evidence of Understanding

  • describe an input in terms of its corresponding output
    • approximate and justify the input of an exponential function given the output
      • Example: given 59 = 2x, students recognize 2xas a continuous function and that 59 is between 32 and 64 and therefore x is a value between 5 and 6, closer to 6
    • describe an exponential function in terms of the output, f(x), and its base
      • Example: f(x) = 2x can be described as “base 2 is f(x) when raised to the exponent x” or “f(x) is equivalent to the base 2 raised to exponent x”
    • use the outputs of an exponential function to graph its inverse function
      • create and analyze a table of values for the inverse function
      • describe the intercepts and end behavior of the inverse function
         
  • create equivalent representations for exponential and logarithmic functions
    • analyze relationships between exponential and logarithmic equations using graphs or tables
      • Example: compare Y= 2x, log2Y= X, and log2X = Y using features from their graphs
      • recognize when equations model the same function, inverse functions, or neither
    • create a table or graph from a simple logarithmic function rule (including natural logarithm)
      • determine the output of a simple logarithm without a calculator (Example: log327 is 3 or log216 is 4 or log418 is a number slightly more than 2)
    • create a simple exponential or logarithmic function rule that best models a given graph or table
    • justify connections between features of the table, graph, and function rule of a given logarithm
       
  • illustrate exponentials and logarithms are inverse functions
    • prove the composition of an exponential function and its inverse log function are commutative
    • prove an exponential function and its inverse log function map onto one another over y = x
      • describe how an exponential function is the reflection of its inverse log function
    • algebraically solve for the inverse function rule for an exponential or log function

Develop conceptual understanding:

inverse, logarithm, natural logarithm

Supporting terms to communicate:

function, input, output, domain, range, exponential, base, e, natural base, intercept, end behavior, asymptote, inverse, commutative, reflect 

Core Resource
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    Instructional Routine: Contemplate then Calculate
    These tasks are embedded within the instructional routine called Contemplate then Calculate. COMING SOON!