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 = 2*^{x}*, students recognize 2*^{x}^{}as 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) = 2*^{x}*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

- approximate and justify the input of an exponential function given the output
**create equivalent representations for exponential and logarithmic functions**- analyze relationships between exponential and
**logarithmic**equations using graphs or tables*Example: compare Y= 2*^{x}*, log*_{2}*Y= X, and log*_{2}*X = 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: log*_{3}*27 is 3 or log*_{2}*16 is 4 or log*_{4}*18 is a number slightly more than 2)*

- determine the output of a simple logarithm without a calculator
- 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

- analyze relationships between exponential and
**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