Goals—•Recognize and evaluate logarithmic functions with base a•Graph Logarithmic functions•Recognize, evaluate, and graph natural logs•Use logarithmic functions to model and solve real-life problems.

Is this function one

to one?

Does it have an inverse?

Must pass the horizontal line

test.

Yes

Yes

f(x) = logax is called the logarithmic function of base a.

Definition: Logarithmic function of base “a” -

For x > 0, a > 0, and a 1,y = logax if and only if x = ay

Read as “log base a of x”

The most important thing to remember about logarithms

is…

a logarithm is an exponent.

Therefore, all logarithms can be written as exponential equations and all exponential equations can be written as logarithmic equations.

log381 = 4 log168 = 3/4

Write the exponential equation in logarithmic form

82 = 64 4-3 = 1/64

34 = 81 163/4 = 8

log 8 64 = 2

log4 (1/64) = -3

f(x) = log42

f(x) = log31

f(x) = log10(1/100)

Step 1- rewrite it as an exponential

equation. 2y = 32Step 2- make the bases

the same. 2y = 25

Therefore,

y = 5

4y = 222y = 21

y = 1/2

3y = 1y = 0

10y = 1/100

10y = 10-2

y = -2

Think: y = log232

f(x) = log232

You can only use a calculator when the base is10

Find the log key on your calculator.

Evaluate the following using that log key.

log 10 = 1

log 1/3 = -.4771

log 2.5 = .3979

log -2 = ERROR!!!

Why?

loga1 = 0 because a0 = 1logaa = 1 because a1 = alogaax = x and alogax = xIf logax = logay, then x = y

log41=

log77 =

6log620 =

Rewrite as an exponent 4y = 1 Therefore, y = 0

Rewrite as an exponent 7y = 7 Therefore, y = 1

0

1

20

log3x = log312

log3(2x + 1) = log3x

log4(x2 - 6) = log4 10

x = 12

2x + 1 = xx = -1

x2 - 6 = 10x2 = 16x = 4

Review: How do you find the inverse of a function?

Application of what you know…What is the inverse of f(x) = 3x?

y = 3x

x = 3y

y = log3xf-1(x) = log3x

Rewrite the exponential as a logarithm…

Find the inverse of the following exponential functions…

f(x) = 2x f-1(x) = log2x f(x) = 2x+1 f-1(x) = log2x - 1

f(x) = 3x- 1 f-1(x) = log3(x + 1)

Find the inverse of the following logarithmic functions…

f(x) = log4x f-1(x) = 4x

f(x) = log2(x - 3) f-1(x) = 2x + 3

f(x) = log3x – 6 f-1(x) = 3x+6

Graphs of Logarithmic Functions

It is the inverse of y = 3x

y = 3x

x y-1 1/3

0 11 32 9

y= log3x

x y1/3 -1

1 03 19 2

Therefore, the table of values for

g(x) will be the reverse of the

table of values for y = 3x.

Domain? (0,)

Range? (-,)

Asymptotes? x = 0

Graphs of Logarithmic Functionsg(x) = log4(x – 3)What is the inverse exponential function?

y= 4x + 3Show your tables of values.

y= 4x + 3x y-1 3.250 41 72 19

y= log4(x – 3)

x y3.25 -1

4 07 1

19 2

Domain? (3,)

Range? (-,)

Asymptotes? x = 3

Graphs of Logarithmic Functionsg(x) = log5(x – 1) + 4What is the inverse exponential function?

y= 5x-4 + 1Show your tables of values.

y= 5x-4 + 1

x y3 1.24 25 66 26

y= log5(x – 1) + 4

x y1.2 32 46 5

26 6

Domain? (1,)

Range? (-,)

Asymptotes? x = 1

The function defined by f(x) = logex = ln x, x > 0

is called the natural logarithmic function.

Find the ln key on your calculator.

Evaluate the following using that ln key.

ln 2 = .6931

ln 7/8 = -.1335ln 10.3 = 2.3321ln -1 = ERROR!!!

Why?

ln1 = 0 because e0 = 1Ln e = 1 because e1 = eln ex = x and eln x = xIf ln x = ln y, then x = y

ln 1/e=

2 ln e =

eln 5=

Rewrite as an exponent ey = 1/e

ey = e-1

Therefore, y = -1Rewrite as an exponent ln e = y/2

e y/2 = e1 Therefore, y/2 = 1 and

y = 2.

-1

2

5

Graphs of Natural Log Functionsg(x) = ln(x + 2)

Show your table of values.

y= ln(x + 2)

x y-2 error-1 00 .6931 1.0992 1.386

Domain? (-2,)

Range? (-,)

Asymptotes? x = -2

Graphs of Natural Log Functionsg(x) = ln(2 - x)

Show your table of values.

y= ln(2 - x)

x y2 error1 00 .693-1 1.099-2 1.386

Domain? (-2,)

Range? (-,)

Asymptotes? x = -2