Logarithm

Logarithm (Introduction)

xbn log xnb

The logarithmic function is defined as the inverse of the exponential

function.  *A LOGARITHM is an exponent.  It is the exponent to which the base must

be raised to produce a given number.

For b > 0 and b 1

is equivalent to  

Number

Exponent

Base  

xnb nxb log

• Examples,

1. Since , then

2. Since , then

3. Since , then

4. Since , then

4162log 4216

4381

8216

3

113

8162log

4813log

13

13log

Properties of Logarithm

1. because

2. because

3. becausexa xa log

1log aa

01log a 10 a

aa 1

xaxa

Rules of Logarithm

1.

2.

3.

NMMN aaa loglog)(log

NMN

Maaa logloglog

MakkMa loglog

• Example:-

1) 810log12510log

31010log331010log100010log812510log

)(log Ma ,ThenY= .Ma y

Now we solve .Ma y For y, using base-b logarithms:

.Ma y

)(log)(log Ma by

b If:

)(log)(log. May bb

)(log

)(log

a

My

b

b

Changing the base:

IF:

Take the base-b logarithm of each side

Power rule

Divide each side by )(log ab

)(log

)(log)(log

ab

MbMa

Base-change formula:

If a and b are positive numbers not equal to 1 and M is positive, then

If the new base is 10 or e, then

)ln(

)ln(

)log(

)log()(log

a

M

a

MMa

Example :

find To four decimal places

• Solution: By using the base-change with a=7 and b=10:

Chick by finding with calculator. Note that we also have

3614.2)7ln(

)99ln()99(log7

)99(log7

3614.27

3614.2)7log(

)99log()99(log7

Common logarithms

• common logarithm is the logarithm with base 10.

• It is indicated by or

sometimes Log(x) with a capital L

• Traditionally, log10 is abbreviated to log.

)(log10 x

• Example :

01log1100

110log10101

2100log100102

110

1log

10

110 1

2100

1log

100

110 2

Binary logarithm

• In mathematics, the binary logarithm is the logarithm for base 2. It is the inverse function of .

• Domain and range: the domain of the exponential function

is and its range is

Because the logarithm function is the inverse of The domain of is and its range is

n2log

n2

xy 2

),(

),0(

)(log 2 xy xy 2log

),0(

),(

xy 2

Examples :

38

1log

8

12 2

3

532log322 25

24

1log

4

12 2

2

12

1log

2

12 2

1

416log162 24

Logarithmic equation :

If we have equality of two logarithms with the same base, we use the one-to-one property to eliminate the logarithm.

If we have an equation with only one logarithm,

such we use the definition of logarithm to write and to eliminate the logarithm

NthenMNaMaif ,loglog

yxa )(log

xa y

• Find the solution :

• Solution:xx 32 )1(

xx 3log)1(2log

)3log(.2log)1( xx

)3log(.)2log()2log(. xx

)2log()3log(.)2log(. xx

)2log()3log()2log( x

)3log()2log(

)2log(

x

xx 32 )1(

Original equation

Take log of each side

Power rule

Distributive property

Get all x-terms on one side

Factor out x

Exact solution

4)3(log)3(log 22 xx

4)3)(3(log2 xx

49log 22 x

42 29 x

252 x

5x

Example (2)

Solve

Solution :

4)3(log)3(log 22 xx

Original equation

Product rule

Multiply the binomials

Definition of logarithm

Even root property

To check, first let x=-5 in the original equation :

Because the domain of any logarithmic function is the set of positive numbers, these logarithms are undefined. Now check x=5 in the original equation :

The solution is {5}.

413

42log8log

4)35(log)35(log

22

22

4)8(log)2(log

4)35(log3)5(log

22

22

Incorrect

Correct

• The natural logarithm is a logarithm to base e

• Where e = 2.7182818….

• it is denoted ln x, as ln x = loge x

Natural Logarithm

Reason for being "natural"

The reason we call the ln(x) "natural" :• expressions in which the unknown variable

appears as the exponent of e occur much more often than exponents of 10

• the natural logarithm can be defined quite easily using a simple integral or Taylor series--which is not true of other logarithms

• there are a number of simple series involving the natural logarithm, and it often arises in nature. Nicholas Mercator first described them as log naturalis before calculus was even conceived.

The general definition of a logarithm

Y = ln x means the same as x = ey

And this leads us directly to the following:

• ln 1 = 0 because e0= 1

• ln e = 1 because e1= e

• ln e2= 2 and ln e-3= -3

Properties:

• All the usual properties of logarithms hold for the natural logarithm, for example:

– (where 28 is an arbitrary real number)

• ln (x)a = a ln x

28

logln

loglog 28

x

xxx

e

4eln 4

Example 1:

1. ln(e)4=

4ln(e) = 4(1)

& since ln e = 1

Example 2:

10ln1

10lnln

10ln

e

e

Example 3:

• ln 5e=

= ln 5 + ln e

= ln ( 5 )+ 1

Example 4:

• It doesn't exist! Why?

?1ln

ln

log1

e

e

Example 5

• e ln 6 = ?

• e ln 6 = 6

Proof that d/dx ln(x) = 1/x

F (x) = ln(x)

1. . f ‘ (x) = lim h-->0 (f (x + h) – f (x)) /h• Definition of a derivative

2. = lim h-->0 (ln(x + h) - ln(x))/h• Plugging the function f (x) = ln(x)

3. = lim h-->0 ln( (x + h) /x) /h• Rule of logarithms: log (a) – log (b) = log (a/b)

4. = lim h-->0 ln(1 + h/x)/h• Algebraic simplification: (x + h)/x = 1 + x/h

5. = lim h-->0 ln(1 + h/x) (⋅ x/h) (1/⋅ x)• Algebraically, 1/h = (x/h)(1/x)

6. = 1/x lim ⋅ h-->0 ln(1 + h/x) (⋅ x/h)• 1/x is a constant with respect to the variable being

"limited," so we can pull it out of the limit .

7. = 1/x lim_⋅ h-->0 ln((1 + h/x)x/h)• Rule of logs: log(a) b = log(ab) ⋅• Let's look at a definition of e using a limit:

• e = lim n-->∞ (1 + 1/n)n Or equivalently: e = lim n-->0 (1 + n)1/n

– lim h-->0 (1 + h/x) x/h = e• True from the definition of e (the x is irrelevant,

since it's constant with respect to h)• 1/x lim_⋅ h-->0 ln((1 + h/x)x/h)

8. = 1/x ln(e)⋅• Follows from (7.5) applied to (7)Since e is

the base of ln: ln (e) = 1

9. = 1/x• What happens when you multiply anything

by 1 is that it doesn't change.

Compare between the graphs ofCompare between the graphs of::

xxf 3)(

xxf 31 log)(

xy

xxf 3)( xxf 31 log)(

)b,1(

b1

1

yxb log

xy blog1) for any base ,x-intercept is 1.because the logarithm of 1 is 0

.2) The graph passes through the point (b,1) .because the logarithm of the base is 1.

1log by b

3) The graph is below the x-axis, the logarithm is negative for

10 xWhich number are those that have negative logarithms.

01log by

Ex:(4)Ex:(4)Graph the followingGraph the following

yx

3-/10001

-21/100

-11/10

01

110

2100

31000

)(log1 xo Sol:Sol:

Change to Change to exponential form,exponential form,

yx 10

Ex:(5)Ex:(5)Graph the followingGraph the following

yx

3-/81

-21/4

-11/2

01

12

24

38

)(log 2 x

yx 2

• Sol:Sol:• Change to Change to

exponential form,exponential form,

Ex:(3)Ex:(3)Graph the followingGraph the following..

Sol:Sol:• Change to Change to

exponential exponential form,form,

YX

-30.04

-20.13

-10.36

01

12.71

27.38

320.08

yex

xy ln

Compare between the Compare between the graphs ofgraphs of::

)(log1 xo

xy ln

EXEX : :

SolveSolve

S0l:S0l:

2)9(log2 x

5

94

94

922

x

x

x

x

No No solutionsolution

Family of Logarithm

1( yxb log

Family of logarithm

2( yxb log

Family of logarithm

3) yxb 3log

Family of logarithm

4) yxb 3log

Family of logarithm

5) yxb 3log

Family of Logarithm

6) yxb 3log