Logarithmic FunctionsLogarithmic FunctionsLogarithmic FunctionsLogarithmic Functions

Logarithm = Exponent

Very simply, a logarithm is an exponent of ten that will produce the desired number.

Y = Log 100 means what is the exponent of 10 which will produce 100.Y = log .001 means what is the exponent of 10 which will produce .001.Note: when no base is indicated, the base is 10.

Y=logb a can be read “ y is the exponent of base b to produce a.”

3= log2 8 is read “3 is the exponent of 2 to produce 8”

2 = log4 16

42 = 16

22

1 12 log 2

4 4

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4

y = log2( )2

1= 2

y2

1

Write the equivalent exponential equation and solve for y.

1 = 5 yy = log51

16 = 4y y = log416

16 = 2yy = log216

SolutionEquivalent Exponential Equation

Logarithmic Equation

16 = 24 y = 4

2

1= 2-1 y = –1

16 = 42 y = 2

1 = 50 y = 0

Definition of a Logarithm

• A logarithm, or log, is defined in terms of an exponent.

• If b x=a, then logb a =x

If 5 2=25 then log5 25=2

Log5 25=2 is read “log base 5 of 25 is 2.”

– You might say the log is the exponent we apply to 5 to make 25

Log of a Product

• The log of a product is the sum of the logs of the factors

logbxy = logbx + logby

Log2512 = log2(8·64)

= log28 + log264

= 3 + 5 = 8

Log of a Quotient

The log of a quotient is the difference of the logs of the factors.

log log logb b bx x yy

Ex. 5 5 5125

log log 125 log 2525

3 2

1

Log of a Power

The log of a power is the product of the exponent and the log of the base.

logbxn = nlogbx

Ex: log 32 = 2log3

Use the properties of logs to simplify the following:

log 2 .3456 log 3 .4234log 5 .4985 log 7 .6002a

a a

a

Given:

34

log 15 log 42

5 2log log

3 7

log 7 log 2 5

a a

a a

a a

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10

For x 0 and 0 a 1, y = loga x if and only if x = a y.The function given by f (x) = loga x is called the logarithmic function with base a.

Every logarithmic equation has an equivalent exponential form: y = loga x is equivalent to x = a y

A logarithmic function is the inverse function of an exponential function.

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11

log10 –4 LOG –4 ENTER ERROR

REMEMBER: no power of 10 gives a negative number

The logarithm function f (x) = log10 x is called the common logarithm function.

log10 100

log10 5

Function Value

Keystrokes

Display

LOG 100 ENTER 2

LOG 5 ENTER 0.69897005

2log10( ) – .3979400LOG ( 2 / 5 ) ENTER

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12

Properties of Logarithms

1. loga 1 = 0 since a0 = 1. 2. loga a = 1 since a1 = a. 3. If loga x = loga y, then x = y. one-to-one property

INVERSE PROPERTIES:

The logarithm with base a of a raised to a power equals that power:

a raised to the logarithm with base a of a number equals that number

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13

log xa a x

loga xa x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 14

One way to Graph f (x) = log2 xSince the logarithm function is the inverse of

the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x.

83422110

–1

–22xx

4

1

2

1

y = log2 x

y = xy = 2x

(1, 0)x

y

x-intercept

horizontal asymptote y = 0

vertical asymptote x = 0

Graphing Log Functions

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 16

One way to Graph f (x) = log2 xSince the logarithm function is the inverse of

the exponential function of the same base, its graph is the reflection of the exponential function in the line y = x.

83422110

–1

–22xx

4

1

2

1

y = log2 x

y = xy = 2x

(1, 0)x

y

x-intercept

horizontal asymptote y = 0

vertical asymptote x = 0

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17

Graph of the common logarithm function f(x) = log10 x.

by calculator

1

10

10.6020.3010–1–2f(x) = log10

x

10421x 1

100

y

x5

–5

f(x) = log10 x

x = 0 vertical asymptote

(0, 1) x-intercept

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18

The graphs of logarithmic functions are similar for different values of a. f(x) = loga x (a 1)

3. x-intercept (1, 0)

5. increasing6. continuous7. one-to-one 8. reflection of y = a

x in y = x

1. domain

),0( 2. range

),(

4. vertical asymptote )(0 as 0 xfxx

Graph of f (x) = loga x (a

1)

x

y y = x

y = log2

x

y = a x

domain

range

y-axisverticalasymptote

x-intercept(1, 0)

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19

Using the graphing calculator to graph the log

functions

• For other than base 10, use the following formula:

10

10

loglog

loga

xx

a

To graph the function

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20

Graph the following functions:

2

3

4

5

( ) log

( ) log

( ) log

( ) log

f x x

f x x

f x x

f x x

2 3log logy x y x

4 5log logy x y x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22

The function defined by f(x) = loge x = ln x

is called the natural logarithm function.

Use a calculator to evaluate: ln 3, ln –2, ln 100

ln 3ln –2ln 100

Function Value

Keystrokes DisplayLN 3 ENTER 1.0986122

ERRORLN –2 ENTERLN 100 ENTER 4.6051701

y = ln x

(x 0, e .718281)

y

x5

–5

y = ln x is equivalent to e y

= x

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23

Properties of Natural Logarithms 1. ln 1 = 0 since e0 = 1. 2. ln e = 1 since e1 = e. 3. ln ex = x and eln x = x inverse property 4. If ln x = ln y, then x = y. one-to-one property

Examples: Simplify each expression.

2

1ln

e 2ln 2 e inverse property

20lne 20 inverse property

eln3 3)1(3 property 2

00 1ln property 1

.

.