L11 Exponential and Logarthmic Functions

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EXPONENTIAL AND

LOGARITHMIC

FUNCTIONS



EXPONENTIAL FUNCTION

If x and b are real numbers such that b > 0 and

b ≠ 1, then f(x) = b x

is an exponential function with base b.

Examples of exponential functions:

a) y = 3x b) f(x) = 6x c) y = 2x

Example: Evaluate the function y = 4 x at the given values

of x. a) x = 2 b) x = -3 c) x = 0



PROPERTIES OF EXPONENTIAL FUNCTION y = b x

• The domain is the set of all real numbers.

• The range is the set of positive real numbers.

• The y – intercept of the graph is 1.• The x – axis is an asymptote of the graph.

• The function is one – to – one.



The graph of the function y = b x

1

o

y

x

axis x: Asymptote Horizontal

none:ercept int x

1 ,0:ercept int y

,0: Range

,: Domain

xb y



EXAMPLE 1: Graph the function y = 3 x

1

X -3 -2 -1 0 1 2 3

y 1/27 1/9 1/3 1 3 9 27

o

y

x

x3 y



EXAMPLE 2: Graph the function y = (1/3) x

1

X -3 -2 -1 0 1 2 3

y 27 9 3 1 1/3 1/9 1/27

o

y

x

x

3

1

y



NATURAL EXPONENTIAL FUNCTION: f(x) = e x

1

o

y

x

axis x: Asymptote Horizontal

none:ercept int x

1 ,0:ercept int y

,0: Range

,: Domain

xe x f



LOGARITHMIC FUNCTION

For all positive real numbers x and b, b≠

1, theinverse of the exponential function y = b x is the

logarithmic function y = logb x.

In symbol, y = logb x if and only if x = by

Examples of logarithmic functions:

a) y = log3 x b) f(x) = log

6 x c) y = log

2 x



EXAMPLE 1: Express in exponential form:

204.0log )d

416log )c

532log ) b

364log )a

5

2

1

2

4

749 )d

8127 )c

3216 ) b

2166 )a

2

1

3

4

4

5

3

EXAMPLE 2: Express in logarithmic form:



PROPERTIES OF LOGARITHMIC FUNCTIONS

• The domain is the set of positive real numbers.

• The range is the set of all real numbers.

• The x – intercept of the graph is 1.• The y – axis is an asymptote of the graph.

• The function is one – to – one.



The graph of the function y = logb x

1o

y

x

axisy:AsymptoteVertical

none:erceptinty

1,0:erceptintx

,:Range

,0:Domain

xlog y b



EXAMPLE 1: Graph the function y = log3 x

1

X 1/27 1/9 1/3 1 3 9 27

y -3 -2 -1 0 1 2 3

o

y

x

xlog y 3



EXAMPLE 2: Graph the function y = log1/3 x

1o

y

x

X 27 9 3 1 1/3 1/9 1/27

y -3 -2 -1 0 1 2 3

xlog y3

1



PROPERTIES OF EXPONENTS

If a and b are positive real numbers, and m and n

are rational numbers, then the following

properties holds true:

mmm

mnnm

nm

n

m

nmnm

baab

aa

aa

a

aaa

mnn mn

m

nn

1

m

m

m

mm

aaa

aa

a

1a

b

a

b

a



To solve exponential equations, the following

property can be used:

bm = bn if and only if m = n and b > 0, b ≠ 1

EXAMPLE 1: Simplify the following:

EXAMPLE 2: Solve for x:

x4xx

2x

5x12x1x24x

273 d) 162

1 )c

84 b) 33 )a

52

10

24

32x ) b

3x )a



PROPERTIES OF LOGARITHMS

If M, N, and b (b ≠ 1) are positive real numbers,

and r is any real number, then

x b

x blog

01log

1 blog

Nlogr Nlog

NlogMlog

N

Mlog

NlogMlogMNlog

xlog

x

b

b

b

b

r

b

b b b

b b b

b



Since logarithmic function is continuous and one-

to-one, every positive real number has a unique

logarithm to the base b. Therefore,

logbN = logbM if and only if N = M

EXAMPLE 1: Express the ff. in expanded form:

24

35

2

5

2

6

3423

t

mnplog )c

py

xlog e) x3log ) b

yxlog d) xyzlog )a



EXAMPLE 2: Express as a single logarithm:

plog3

2nlog2mlog32log )c

nlog3mlog2 ) b

3logxlog2xlog a)

5555

aa

222



NATURAL LOGARITHM

Natural logarithms are to the base e, while

common logarithms are to the base 10. Thesymbol ln x is used for natural logarithms.

2ln3xlnlne a) ln x

EXAMPLE: Solve for x:

1elogeln

xlogxln

e

e



CHANGE-OF-BASE FORMULA

0.1log c)

70log b)

65log a)

2

0.8

5

EXAMPLE: Use common logarithms and natural

logarithms to find each logarithm:

bln

ln x xlog or

blog

xlog xlog b

a

a b



Solving Exponential Equations

Guidelines:

1. Isolate the exponential expression on one side ofthe equation.

2. Take the logarithm of each side, then use the law of

logarithm to bring down the exponent.

3. Solve for the variable.


06ee )d

4e )c

20e8 ) b

73 )a

xx2

x23

x2

2x

S l i i h i i



Solving Logarithmic Equations

Guidelines:

1. Isolate the logarithmic term on one side of the

equation; you may first need to combine thelogarithmic terms.

2. Write the equation in exponential form.

3. Solve for the variable.

EXAMPLE 1: Solve the following:

2x264

9log )d

2

5xlog ) b

4

x

25

4

log )c 327

8

log )a

8

34

52x




11xlog5xlog )f

xlog2xlog6xlog )e

25xlog25xlog d)

8ln x)c

3x25log b)

162xlog34 a)

77

222

52

5

2

li i ( l d h )



Application: (Exponential and Logarithmic Equations)

1. The growth rate for a particular bacterial culture can be

calculated using the formula B = 900(2)t/50, where B is

the number of bacteria and t is the elapsed time inhours. How many bacteria will be present after 5 hours?

2. How many hours will it take for there to be 18,000

bacteria present in the culture in example (1)?

3. A fossil that originally contained 100 mg of carbon-14

now contains 75 mg of the isotope. Determine the

approximate age of the fossil, to the nearest 100 years, if

the half-life of carbon-14 is 5,570 years.

isotopetheof lifeHalf k

presentisotopeof amt.orig.reducetoit takestime t

isotopeof amt..origA

isotopeof amt. presentA : where2AA

o

k

t

o

4 I f 15 000 l h d f h



4. In a town of 15,000 people, the spread of a rumor that

the local transit company would go on strike was such

that t hours after the rumor started, f(t) persons heard

the rumor, where experience over time has shown that

a) How many people started the rumor?

b) How many people heard the rumor after 5 hours?5. A sum of $5,000 is invested at an interest rate of 5% per

year. Find the time required for the money to double if

the interest is compounded (a) semi-annually (b)

continuously.

t8.0e74991

000,15tf

lycontinuouscompoundederestintPetA

year pern timescompounded erestintn

r 1PtA

year 1forerestintsimpler 1PA

tr

tn

L11 Exponential and Logarthmic Functions

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