EXPONENTIAL AND LOGARITHMIC FUNCTIONS

EXPONENTIAL FUNCTIONIf x and b are real numbers such that b > 0 and

b ≠ 1, then f(x) = bx 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 = 4x at the given values of x.

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

PROPERTIES OF EXPONENTIAL FUNCTION y = bx

• 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 = bx

1

o

y

x

axisx:Asymptote Horizontal

none:erceptintx

1,0:erceptinty

,0:Range

,:Domain

xby

EXAMPLE 1: Graph the function y = 3x

1

X -3 -2 -1 0 1 2 3

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

o

y

x

x3y

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

1y

NATURAL EXPONENTIAL FUNCTION: f(x) = ex

1

o

y

x

axisx:Asymptote Horizontal

none:erceptintx

1,0:erceptinty

,0:Range

,:Domain

xexf

LOGARITHMIC FUNCTIONFor all positive real numbers x and b, b ≠ 1, the

inverse of the exponential function y = bx 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) = log6 x c) y = log2 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:Asymptote Vertical

none:erceptinty

1,0:erceptintx

,:Range

,0:Domain

xlogy 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

xlogy 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

xlogy3

1

PROPERTIES OF EXPONENTSIf a and b are positive real numbers, and m and n are rational numbers, then the following properties holds true:

mmm

mnnm

nmn

m

nmnm

baab

aa

aa

a

aaa

mnn mn

m

nn

1

mm

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:

x4xx2x

5x12x1x24x

273 d) 162

1 )c

84 b) 33 )a

5

210

24

32x )b

3x )a

PROPERTIES OF LOGARITHMSIf M, N, and b (b ≠ 1) are positive real numbers, and r is any real number, then

xb

xblog

01log

1blog

NlogrNlog

NlogMlogN

Mlog

NlogMlogMNlog

xlog

xb

b

b

br

b

bbb

bbb

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

52

6

3423

t

mnplog )c

py

xlog e) x3log )b

yx log d) xyz log )a

EXAMPLE 2: Express as a single logarithm:

plog3

2nlog2mlog32log )c

nlog3mlog2 )b

3logxlog2xlog a)

5555

aa

222

NATURAL LOGARITHMNatural logarithms are to the base e, while

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

2ln3xlnlne a) ln x EXAMPLE: Solve for x:

1 elog e ln

x log x ln

e

e

CHANGE-OF-BASE FORMULA

0.1 log c)

70 log b)

65 log a)

2

0.8

5

EXAMPLE: Use common logarithms and natural logarithms to find each logarithm:

bln

ln x x log or

b log

xlog x log b

a

ab

Solving Exponential EquationsGuidelines:1. Isolate the exponential expression on one side of the

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

logarithm to bring down the exponent.3. Solve for the variable.

EXAMPLE: Solve for x:

06ee )d

4e )c

20e8 )b

73 )a

xx2

x23

x2

2x

Solving Logarithmic EquationsGuidelines:1. Isolate the logarithmic term on one side of the

equation; you may first need to combine the logarithmic 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

4log )c 3

27

8log )a

8

34

5

2x

EXAMPLE: Solve for x:

11xlog5xlog )f

xlog2xlog6xlog )e

25xlog25xlog d)

8ln x )c

3x25log b)

16 2x log 34 a)

77

222

52

5

2

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 in hours. How many bacteria will be present after 5 hours?

2. How many hour 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.

isotope theof lifeHalfk

present isotope of amt. orig. reduce toit takes time t

isotope of amt. .origA

isotope of amt.present A : where2AA

o

k

t

o

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

lycontinuous compounded erestintPetA

yearper n times compounded erestintn

r1PtA

year 1for erestint simpler1PA

t r

t n