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Georgia Standards

These are the resources that support this Georgia Standard.

A.FGR.9.5 - Compare characteristics of two functions each represented in a different way.

Displaying 101 - 150 of 183 resources:

Thumbnail Image Title Description Curriculum Nodes
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 04, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 04, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 04, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 04, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 04, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 04, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 04, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 04, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 04, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 05, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 05, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 05, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 05, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 05, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 05, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 05, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 05, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 05, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 06, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 06, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 06, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 06, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 06, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 06, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 06, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 06, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 06, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 07, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 07, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 07, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 07, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 07, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 07, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 07, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 07, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 07, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 08, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 08, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 08, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 08, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 08, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 08, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 08, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 08, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 08, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 09, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 09, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 09, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 09, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 09, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 09, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 09, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 09, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 09, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 10, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 10, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 10, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 10, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 10, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 10, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 10, Quiz 10, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 10, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 10, Quiz 10, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 01, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 01, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 01, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 01, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 01, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 01, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 01, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 01, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 01, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 02, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 02, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 02, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 02, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 02, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 02, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 02, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 02, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 02, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 03, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 03, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 03, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 03, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 03, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 03, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 03, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 03, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 03, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 04, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 04, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 04, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 04, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 04, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 04, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 04, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 04, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 04, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 05, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 05, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 05, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 05, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 05, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 05, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 05, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 05, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 05, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 06, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 06, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 06, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 06, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 06, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 06, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 06, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 06, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 06, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 07, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 07, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 07, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 07, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 07, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 07, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 07, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 07, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 07, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 08, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 08, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 08, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 08, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 08, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 08, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 08, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 08, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 08, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 09, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 09, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 09, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 09, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 09, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 09, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 09, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 09, Level 3

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 09, Level 3

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 10, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 10, Level 1

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 10, Level 1

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions
Paper-and-Pencil Quiz--Graphs of Exponential Functions, Base 2, Quiz 10, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 10, Level 2

Paper-and-Pencil Quiz: Graphs of Exponential Functions, Base 2, Quiz 10, Level 2

This is part of a collection of math quizzes on the topic of graphs of exponential functions.

Exponential and Logarithmic Functions and Equations and Graphs of Exponential and Logarithmic Functions