logarithmic functions(1) (1)

7/31/2019 Logarithmic Functions(1) (1)

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Logarithmic Functions


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In this section, another type of function will be studied called

the logarithmic function. There is a close connection

between a logarithmic function and an exponential function.

We will see that the logarithmic function and exponentialfunctions are inverse functions.


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The logarithmic function with base two is defined to be the

inverse of the one to one exponential function

Notice that the exponential

function

is one to one and therefore has

an inverse.

0

1

2

3

4

5

6

7

8

9

-4 -2 0 2 4

graph of y = 2^(x)

approaches the negative x-axis as x gets

large

passes through (0,1)

2

x

y

2x

y


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Inverse of an Exponential Function

Start with

Now, interchangex andy coordinates:

There are no algebraic techniques that can be used to solve for

y, so we simply call this functiony the logarithmic function

with base 2. The definition of this new function is:

if and only if

2xy

2yx

2log x y 2yx


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Logarithmic-Exponential

Conversions

Study the examples below. You should be able to convert a

logarithmic into an exponential expression and vice versa.

1.

2.

3.

4.

4log (16) 4 16 2xx x

3125 5 5log 125 3

1

281

181 9 81 9 log 9

2

3

3 3 33

1 1log ( ) log ( ) log (3 ) 3

27 3


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Solving Equations

Using the definition of a logarithm, you can solve equations

involving logarithms. Examples:

3 3 3log (1000) 3 1000 10 10b b b b

56log 5 6 7776x x x

In each of the above, we converted from log form to

exponential form and solved the resulting equation.


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Properties of Logarithms

These are the properties of logarithms.MandNare positive real

numbers, b not equal to 1, andp andx are real numbers.(For 4, we needx > 0).

5. log log log

6. log log log

7. log log

8. log log

b b b

b b b

p

b b

b b

MN M N

MM N

N

M p M

M N iff M N

log

1.log (1) 02.log ( ) 1

3.log

4. b

b

b

x

b

x

b

b x

b x


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Solving Logarithmic Equations

1. Solve forx:


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Solving Logarithmic Equations

1. Solve forx:

2. Product rule

3. Special product

4. Definition of log

5. x can be +10 only

6. Why?

4 4

4

2

4

3 2

2

2

log ( 6) log ( 6) 3

log ( 6)( 6) 3

log 36 34 36

64 36

100

10

10

x x

x x

xx

x

x

x

x


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Another Example

1. Solve:


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Another Example

1. Solve:

2. Quotient rule

3. Simplify

(divide out common factor )

4. rewrite

5 definition of logarithm

6. Property of exponentials


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Common Logs and Natural Logs

Common log Natural log

10log logx x ln( ) logex x

2.7181828e If no base is indicated,the logarithm is

assumed to be base 10.


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Solving a Logarithmic Equation

Solve forx. Obtain the exact

solution of this equation in terms

of e (2.71828)

ln (x + 1)lnx = 1


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Solving a Logarithmic Equation

Solve forx. Obtain the exact

solution of this equation in terms

of e (2.71828)

Quotient property of logs

Definition of (natural log)

Multiply both sides byx

Collectx terms on left side

Factor out common factorSolve forx

ln (x + 1)lnx = 1

xe =x + 1

xe - x = 1x (e-1) = 1

1

1x

e

x

xe

11

11

ln

x

x


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Application

How long will it take money to double

if compounded monthly at 4 %

interest?


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Application

How long will it take money to double

if compounded monthly at 4 %

interest?

1. Compound interest formula

2. ReplaceA by 2P (double the

amount)

3. Substitute values for r and m

4. Divide both sides by P

5. Take ln of both sides6. Property of logarithms

7. Solve for tand evaluate expression

Solution:

12

12

12

1

0.042 1

12

2 (1.003333...)

ln 2 ln (1.003333...)

ln 2 12 ln(1.00333...)

ln 217.36

12ln(1.00333...)

mt

t

t

t

rA P

m

P P

t

t t


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Logarithmic functions compared to

others

Among increasing functions, the logarithmic functions

with bases b > 1 increase much more slowly for largevalues ofx than either exponential or polynomial

functions. When a visual inspection of the plot of a

data set indicates a slowly increasing function, a

logarithmic function ofter provides a good model.


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Comparing functions

20

beat

beat

Logarithmic functions

logarithmic functions(1) (1)

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