4.2 Logarithmic Functions Think back to “inverse functions”. How would you find the inverse of an exponential function such as:

Or more generally:

( ) 2xf x

( ) xf x b

( ) 2

( ) 2

( ) for x 0

f x x

f x x

f x x

Definition of a Logarithmic Function

• For x > 0 and b > 0, b = 1,• y = logb x is equivalent to by = x.

(Notice that this is the INVERSE of the exponential function f(x) = y = bx)

• The function f (x) = logb x is the logarithmic function with base b.

Location of Base and Exponent in Exponential and Logarithmic Forms

Logarithmic form: y = logb x Exponential Form: by = x. Logarithmic form: y = logb x Exponential Form: by = x.

Exponent Exponent

Base Base

To convert from log to exponential form, start with the base, b, and move clockwise across the = sign:

b to the y = x.

Text ExampleWrite each equation in its equivalent exponential form.a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y

Solution With the fact that y = logb x means by = x,

c. log3 7 = y or y = log3 7 means: .

a. 2 = log5 x means:

.

b. 3 = logb 64 means:

Evaluatea. log2 16 b. log3 9 c. log25 5

Solution

log25 5 = ____ because

25___ = 5.

25 to what power is 5?

c. log25 5

log3 9 = ____ because

3__ = 9.

3 to what power is 9?b. log3 9

log2 16 = ____ because

2__ = 16.

2 to what power is 16?

a. log2 16

Logarithmic Expression EvaluatedQuestion Needed for Evaluation

Logarithmic Expression

Basic Logarithmic Properties Involving One

• logb b = because ____is the exponent to which b must be raised to obtain b. (b__ = b).

• logb 1 = because ____ is the exponent to which b must be raised to obtain 1. (b__ = 1).

Inverse Properties of LogarithmsFor b>0 and b 1,

logb bx = x The logarithm with base b of b raised to a power equals that power.

b logb x = x b raised to the logarithm with base b of a number equals that number.

That is: since logarithmic and exponential functions are inverse functions, if they have the SAME BASE they “cancel each other out”.

Properties of LogarithmsGeneral Properties Common

Logarithms*

1. logb 1 = 0 1. log 1 = 0

2. logb b = 1 2. log 10 = 1

3. logb bx = x 3. log 10x = x4. b logb x = x 4. 10 log x = x

* If no base is written for a log, base 10 is assumed. If it says ln, that means the “natural log” and the base is understood to be e.

Natural Logarithms*

1. ln 1 = 02. ln e = 13. ln ex = x4. e ln x = x

Ex:

log 4 4 =

log 8 1 =

3 log 3 6 =

log 5 5 3 =

2 log 2 7 =

ln e =

ln 1 =

e ln 6 =

ln e 3 =

Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.

Solution We first set up a table of coordinates for f (x) = 2x. Reversing these coordinates gives the coordinates for the inverse function, g(x) = log2 x.

4

2

8211/21/4f (x) = 2x

310-1-2x

2

4

310-1-2g(x) = log2 x

8211/21/4x

Reverse coordinates.

Text Example

Graph f (x) = 2x and g(x) = log2 x in the same rectangular coordinate system.

Solution

We now plot the ordered pairs in both tables, connecting them with smooth curves. The graph of the inverse can also be drawn by reflecting the graph of f (x) = 2x over the line y = x.

-2 -1

6

2 3 4 5

5

4

3

2

-1

-2

6

f (x) = 2x

f (x) = log2 x

y = x

Continued…

Where is the asymptote for the exponential function, where is it now for the log function?

What happened to the exponential functions y-intercept?

Characteristics of the Graphs of Logarithmic Functions of the Form f(x) = logbx

• The x-intercept is 1. There is no y-intercept.

• The y-axis is a vertical asymptote.

• If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.

• The graph is smooth and continuous. It has no sharp corners or edges.

Log Graphs using Transformationsshift c up c>0shift c down c<0

shift c right c>0shift c left c<0

reflect about y-axis

reflect about x-axis

( ) log

( ) log ( )

( ) log ( )

( ) log

b

b

b

b

g x x c

g x x c

g x x

g x x

Also do p 421 # 18, 64, 78, 82, 102, if time 119

18. Write in equivalent log form: 3 343b

64. Graph f(x) = logx, then use transformations to graph g(x)= 2-logx. Find the asymptote(s), domain, range, and x- and y- intercepts.

78. Find the domain of f(x) = log (7-x)

82. Evaluate without a calculator: log 1000.

102. Write in exponential form and solve:

5log ( 4) 2x