Warm UpFind the inverses for the following

functions:

Use algebra to determine if the following functions are inverses of each other (no graphing, no tables)…

Sec 6.2.1

LOGARITHMS Pt. 1

Learning TargetsFinding the inverse of an exponential

function

Investigating the Logarithmic Function

How to use the Logarithmic Function

Known inverses:LinearQuadraticCubic term Absolute ValueHyperbola

WHATS MISSING?!?EXPONENTIAL FUNCTIONS

Graph the Inverse of the Following Function

Use a table and the definition of an inverse…-2 1/9 1/9 -2-1 1/3 1/3 -10 1 1 01 3 3 12 9 9 2

-14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-9-8

-7-6

-5

-4

-3

-2

-1

1

2

3

45

678

9

x

y

(-2,1/ 9)(-1,1/ 3)(0,1)

(1,3)

(2,9)

(1/ 9,-2)(1/ 3,-1)

(1,0)(3,1)

(9,2)

What do you notice about it…List of Characteristics:

Asymptote of Grows very slowly…Opposite of our exponential!!!

We call this function a Logarithmic FunctionIt “undoes” an exponential function when the

independent variable is the exponent

-14-13-12-11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 1213 14 15

-9-8-7-6-5-4-3-2-1

123456789

x

y

Before we go too deep into the inverse lets understand Log(arithm)P. 278: 6-56Mathematicians first created a puzzle in

ancient India in the 2nd Century BCE. More recently, about 700 years ago, Muslim mathematicians created the first tables allowing them to find answers to this type of puzzle quickly.

Here are some clues to help you figure out how the puzzle works:

Use these clues to solve for a-f

Log NotationWhat do these clues tell us?

Log form is equivalent the exponential form

Ex:

BASE

EXPONENT

Practice6-68 Modified: Fill in the table.Try looking at your table in the calculator!!!

1/10

1/2 1 2 3 4 5 10 100

1/10

1/2 1 2 3 4 5 10 100

-1 -.30

0 .301

.47 .60 .70 1 2

𝒍𝒐𝒈 𝒙=𝒈 (𝒙)

What base is this function?

Practice6-68 Modified: Fill in the table.

1/5 1/2 1 2 3 4 5 10 100

𝐥𝐨𝐠𝟓𝒙=𝒈 (𝒙)

NotationThe base for is always going to be in base

10.The only time this changes is when the

base is stated.

How to use in calculator:

MATH logBASENow you can find the outputs for any base!!!

Practice6-68 Modified: Fill in the table.

1/5 1/2 1 2 3 4 5 10 100

𝐥𝐨𝐠𝟓𝒙=𝒈 (𝒙)

Log ValuesWhat inputs are not valid for Logs? Why?

Pause and Ponder SILENTLY to come up with a reason…

Oh yea that’s right our exponential function will never give us negative output’s (asymptote)! Therefore we cannot have negative inputs for inverse!

Remember an inverse function “undoes” the original function

Original: XY Inverse:YX

Back to Inverses of Log’sFind the inverse for:(Hint: write it in log form)

You should have the function:

Remember we switch the x and y…

You try…Do problem 6-70 a-d:

Remember to always check your answers….

Lets solve the following…

Your Inverse ToolkitYou should be able to go through your notes and

find the following information:Define what an inverse is and when a function will

have one.Find a functions inverse by “undoing” Prove if two functions are inverses by using:

TableGraphComposition

Restrict the Domain for non-invertible functions in order to come up with an inverse

Use Log’s to solve and invert exponential functions that has an input unknown for the exponent

On your ownReview your notes

and seek help on subjects that are unclear or need reinforcing.

Please fill out an index card that answers the following:

During this unit the topic that is most unclear up to this point is? Why?

Homework6-69, 6-71, 6-73-6-

776-87-6-91