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Inverses

Site: Clare
Course: Michigan Algebra I Sept. 2012
Book: Inverses
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Date: Sunday, November 24, 2024, 12:55 AM

Description

Inverses

Table of contents

Introduction

Just like all families of functions, exponential functions have an inverse. To find the inverse, switch the x and y values of the graph, table or equation to see the new function.

Example Graph the inverse of the function: y = 2x.

Step 1. Make a table of values.

ExpInverseEx1-1

Step 2. Switch the values of x and y in the table.

ExpInverseEx1-2

Step 3. Plot the points on a coordinate plane.

ExpInverseEx1-3

*Note: The asymptote of the original function is y = 0, while the asymptote of the inverse is x = 0. Also, recall from earlier study of inverses, graphs of the inverses are reflected across the line y = x.

Logarithms

The graph in the previous example can be modeled by the function y = log2 x. Logarithms are the inverse of exponentials, just as subtraction is the inverse of addition, and division is the inverse of multiplication. Translating exponential functions into logarithmic functions always follows the form below:

y = bx is equivalent to logb( y) = x

The right-hand side above is pronounced, "log-base-b of y equals x". The value of the subscripted "b" is the base of the logarithm, just as b is the base in the exponential equation. And, just as the base b of an exponential is always positive and never equal to 1, so is the base b for a logarithm.

* Note: Finding the logarithm that corresponds to the exponential function will be covered in Algebra II. Look at the following examples as a way to introduce yourself to the idea that will be covered later.


Examples

Example 1

Convert 63 = 216 to the equivalent logarithmic equation.

To convert, the base remains the same, but the 3 and the 216 switch sides.

log6 (216) = 3

Example 2

Convert log4(1024) = 5 to the equivalent exponential equation.

To convert, the base remains the same, but the 1024 and the 5 switch sides.

4 5 = 1024

Video Lesson

To learn more about logarithms, watch the following video:

Introduction to Logarithms

Practice

Inverses of Exponentials Worksheet

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Answer Key

Inverses of Exponentials Answer Key

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Sources

Khan, Salman. " An introduction to logarithms."

http://www.khanacademy.org/video/introduction-to-logarithms (accessed September 12, 2010).

Stapel, Elizabeth. "Logarithms: Introduction to 'The Relationship'." Purplemath. Available from http://www.purplemath.com/modules/logs.htm. Accessed 12 September 2010