S T U D E N T S W I L L B E A B L E T O :• CONVERT BETWEEN EXPONENT AND LOG FORMS• SOLVE LOG EQUATIONS OF FORM LOGBY=X FOR B, Y, AND X

LOGARITHMIC FUNCTIONS

LOGARITHMS

• For , , and ,if and only if

• The function given by

is called the logarithmic function with base

Read as “log base a

of x”

LOGARITHMS ARE EXPONENTS!

• “What must I raise to in order to get ?• because 2 must be raised to the 3rd power to get 8.

• Why? Logs are a useful way of writing exponents that we don’t know. For example: What must you raise 3 to in order to get 42?

• The log is just a way of writing the exponent!

• Converting between exponential and log form:

CONVERTING BETWEEN LOG AND EXPONENTIAL FORM

•

•

WITHOUT A CALCULATOR, FIND THE FOLLOWING

1.

2.

3.

4.

5.

6.

7.

8.

COMMON LOGARITHMS

• A log in base 10 is called a common logarithm.

• We can write as

• The LOG button on your calculator evaluates common logs only

• for and

PROPERTIES OF LOGS

1. because

2. because

3. and (they are inverse functions)

4. If , then (one to one)

USING THE PROPERTIES OF LOGS

Solve for x:1. 2.

Simplify:1. 2.

HOMEWORK

•Pg 195 # 3-6, 11-20, 23-30•Difference Quotient Video Project due next class!!

GRAPHS OF LOGARITHMIC FUNCTIONS

• Graph

• is the inverse function of • To graph: reflect the graph of over the

line (flip the x and y)

PROPERTIES OF LOG FUNCTIONS:

Domain: (0, ∞)Range: (- ∞, ∞)x-intercept: (1, 0)y-int: noneIncreasing: (0, ∞)Vertical Asymptote: x=0HA: noneContinuous: YesReflection of the graph y = ax in the line y=x f(x)=log x

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

f(x)=log x

f(x)=10^x

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

f(x)=log x

f(x)=10^x

f(x)=x

-4 -3 -2 -1 1 2 3 4

-4

-3

-2

-1

1

2

3

4

x

y

STEPS TO GRAPHING LOG FUNCTIONS

• Identify the base of the log and write it as an exponential function

• Make a table of values

• Flip the x and y values to get log function (the inverse)

• Now apply translations:

• Inside parenthesis: change values• Divide by

• Add (remember, if it’s you should subtract 2)

• Outside parenthesis: change values• Multiply by

• Add

TRANSFORMATIONS OF LOG FUNCTIONS

𝑔 (𝑥)=log2 (𝑥−2)

x y

-2

-1

0

1

2

x y x y

2𝑥 log 2𝑥 log 2(𝑥−2)

Domain:

Range:

VA:

x-int:

TRANSFORMATIONS OF LOG FUNCTIONS

𝑔 (𝑥 )=−3 log 4(𝑥+1)x y

-2

-1

0

1

2

x y x y

Domain:

Range:

VA:

x y

TRANSFORMATIONS OF LOG FUNCTIONS

Domain:

Range:

VA:

x-int:

x y

-2

-1

0

1

2

x y x y x y

FIND THE DOMAIN

1. f(x) = log7(x-4)

2. g(x)= log (1-2x)

3. g(x)= log8 x2

How do you think we

can find the domain?

Remember, we can’t take the

log of a negative number.

Set the piece inside the log greater than or equal to zero, then

solve!

• Since we can choose any ‘c’ we want, we’ll choose base 10 (the common log) which the calculator can do

• Ex1: Find

• Ex2: Find

CHANGE OF BASE FORMULA

Any number you want!

THE NATURAL LOGARITHM

• Base • → read “el en of x” or “the

natural log of x”• Inverse of • Definition for Natural Log:

For if and only if

• Use LN button on your calculator to evaluate.

PROPERTIES OF NATURAL LOGS

1. because

2. because

3. and

4. If , then

USING THE PROPERTIES OF NATURAL LOGS

1.

2.

3.

4.

QUIZ 3.1-3.2 THURSDAY

• 3.1 Exponential Functions and their graphs• Evaluate exponential functions• Graph exponential functions• Properties of exp graphs• Natural base• Graph • Evaluate

• Compound Interest• Solve 1-to1 exp equations

• 3.2 Logarithmic Functions and their graphs• Evaluate log functions• Graph log functions• Properties of log graphs• Properties of logs• Common Log• Change of Base• Natural Log • Evaluate

HOMEWORK

• Pg 195 #35, 41, 43, 45-48, 50, 51(53-61 odd)