Exponential functionExponential function

☺Definition of exponential function…Definition of exponential function…☺Exponents Basic Rules…Exponents Basic Rules…☺Properties of Exponents…Properties of Exponents…☺Exponential function and theirExponential function and their graphs…graphs…☺Application on Exponential function…

What the Exponents areWhat the Exponents are??

They are shorthand for multiplication process…

Examples:Examples:

• 32 =3*3 =9• 57 =5*5*5*5*5*5*5 =78125• 24 =2*2*2*2 =16

Definition of exponentialDefinition of exponential function function::

A function in which the baseA function in which the base““a” is raised to some power…a” is raised to some power…

BaseBase 3

4 powerpower

aa00 =1 =1ExamplesExamples::

60 = 1(0.25)0 = 1

You tryYou try::

(200000000)0= 1

11//aaxx = a = a-x-x

ExamplesExamples::1(/0.5)x = )0.5( –x

It means that when we have fraction and it’s denominator is raised to any power we just move the denominator to the nominator and change the sign of the power

You tryYou try: :

1/4x = 4-x

nn√ a = a√ a = a)1/n()1/n(

Example:Example:2√ 0.5 = 0.5)1/2(

You try:You try:5√ 2 = 2 )1/5(

nn√ a = a√ a = a)1/n()1/n(

Example:4√ 33 = 3 )3/4(

2√ )0.5(4 = )0.5( )4/2(

You try:You try:

5 √ 4559 =

455)9/5(

If the multiplied bases are similar If the multiplied bases are similar we just add the exponent we just add the exponent ”power”….”power”….

aann . a . amm = a = an+mn+m

Example:Example:24*22 = 2)4+2( = 26 = 64

You try:You try:32*35= 21872187

If the Power is raised to a power, Multiply the exponents

((nmnm))aa = = (aann((mm

ExampleExample::(52)2 =5(2*2) = 625

You tryYou try::(23)2=

26 =64

Dividing like bases….

aann/a/amm=a=a)n-m()n-m(

Examples:Examples:25/24= 2(5-4( = 2

24/25= 2(-1( = (0.5(

You try:You try:44/46=

4(4-6( = 4(-2( =1/42 =1/16

Any fraction raised to any power….Any fraction raised to any power….

(33/43 = (27/64

ExampleExample::(2/4(2( = 23/42 = (8/16½=

TryTry::(3/4(3=

((a/ba/b))nn=)a=)ann(/)b(/)bnn((

Y=Y=22xx

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

Domain f= R

Range f= (0, ∞(

Y=2x -2

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

Domain f= R

Range f= (-2, ∞(

Y=2x +2

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

Y=2(x -2(

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

2(x +2(

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

Y=-2x

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

Y=2-x

1 2 3 4-1-2-3-4

x1

2

3

4

-1

-2

-3

-4

y

Application on Exponential function

Exponential functions are useful in modeling many phenomena involving rapid growth. In particular, population growth under ideal conditions can be modeled effectively by exponential functions. Let p(t( represent the size of a population at time (t( . The basic exponential growth model is given by the following equation where d is the doubling time and p is the initial population:

Example:Example: If a bacteria colony initially contains 100 bacteria and the population size doubles every 5 hours, what is the size of the population after 10 hours?

SolutionSolution::The size of the population after 10 hours is:

P(10(= 2(10/5(

100 = 2(2(

100 =400 bacteria

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