9.1 E

XPONENTI

AL

FUNCTI

ONS

EXPONENTIAL FUNCTIONS

A function of the form y=abx,

where a=0, b>0 and b=1.Characteristics

1. continuous and one-to-one2. domain is the set of all real numbers 3. Range is either all real positive numbers or all real negative numbers depending on whether a is < or > 04. x-axis is a horizontal asymptote5.y-intercept is at a6. y=abx and y=a(1/b)x are reflections across the y-axis

EXAMPLE 1

Sketch the graph of y=2x. State the domain and range.

EXAMPLE 2

Sketch y=( )x. State the domain and range.

EXPONENTIAL GROWTH & DECAY

Exponential Growth:

Exponential function with base greater than one. y=2(3x)

Exponential Decay:

Exponential function with base between 0 and 1 y=4(1/3)x

EXAMPLE 3-6

Determine if each function is exponential growth or decay

y=(1/5)x y=7(1.2)x

y=2(5)x y=10(4/3)x

STEPS TO WRITE AN EXPONENTIAL FUNCTION 1. Use the y-intercept to find a

2. Choose a second point on the graph to substitute into the equation for x and y. Solve for b.

3. Write your equation in terms of y=abx (plug in a and b)

EXAMPLE 7

Write an exponential function using the points (0, 3) and (-1, 6)

EXAMPLE 8

Write an exponential function using the points (0, -18) and (-2, -2)

EXAMPLE 9

In 2000, the population of Phoenix was 1,321,045 and it increased to 1,331,391 in 2004.

A. Write an exponential function of the form y=abx that could be used to model the population y of Phoenix. Write the function in terms of x, the number of years since 2000.

B. Suppose the population of Phoenix continues to increase at the same rate. Estimate the population in 2015.

EXPONENTIAL EQUATIONS

Exponential equation:

An equation in which the variables are exponents

Property of EqualityIf the base is a number other than 1 and the base is the same , then the two exponents equal each other.

2x = 28 then x=8

STEPS TO SOLVE EXPONENTIAL EQUATIONS/INEQUALITIES1. Rewrite the equation so all terms have like

bases (you may need to use negative exponents)

2. Set the exponents equal to each other

3. Solve

4. Plug x back in to the original equation to make sure the answer works

EXAMPLE 10

Solve 32n+1 = 81

EXAMPLE 11

Solve 35x = 92x-1

EXAMPLE 12

Solve 42x = 8x-1

EXAMPLE 13

Solve 256

14 13 p

EXAMPLE 14

Solve 1255 32 x

EXAMPLE 15

Solveaa 164 64

9.2 LO

GARITHMS A

ND

LOGARIT

HMIC F

UNCTIONS

Logarithms with base b

Say: “Log base b of x equals y.”

yxb log

LOGARITHMIC TO EXPONENTIAL FORM

216log.2 4

327

1log.3

01log.1

3

8

EXPONENTIAL TO LOGARITHMIC FORM

39.6

1000.4

2

1

3

a 322.5 5

EVALUATE LOGARITHMIC EXPRESSIONS

64log.7 2 81log.8 3

CHARACTERISTICS OF LOGARITHMIC FUNCTIONS

1. Inverse of the exponential function y=bx

2.Continous and one-to-one

3. Domain is all positive real numbers and range is ARN

4. y-axis is an asymptote

5. Contains (1,0), so x-intercept is 1

HELPFUL HINT

Since exponential and logarithmic functions are inverses if the bases are the same they “undo” each other…

143

86log

)14(log

86

3

xx

LOGARITHMIC EQUATIONS

Property of Equality If b is a positive number other than 1, then if and only if x = y.

yx bb loglog

3

3loglog 77

x

x

EXAMPLE 9

Solve2

5log4 n

EXAMPLE 10

Solve )34(loglog 42

4 xx

EXAMPLE 11

Solve pp 5

25 log)2(log

LOGARITHMIC TO EXPONENTIAL INEQUALITY

3

2

2

3log

x

x5

3

30

5log

x

x

If b > 1, x > 0 and logbx > y then x > by

If b > 1, x > 0 and logbx < y then 0< x < by

EXAMPLE 12

Solve 2log5 x

EXAMPLE 13

Solve 3log4 x

PROPERTY OF INEQUALITY FOR LOGARITHMIC FUNCTIONS

If b>1, then if and only if x>y

and if and only if x<y

yx bb loglog

yx bb loglog

EXAMPLE 14

)6(log)43(log 1010 xx

EXAMPLE 15

)5(log)82(log 77 xx

9.3 P

ROPERT

IES O

F

LOGARIT

HMS

PRODUCT PROPERTY

The logarithm of a product is the sum of the logarithm of its factors

nmnm bbb loglog))((log

QUOTIENT PROPERTY

The logarithm of a quotient is the difference of the logarithms of the numerator and denominator.

nmn

mbbb logloglog

POWER PROPERTY

The logarithm of a power is the product of the logarithm and the exponent

mpm bp

b loglog

EXAMPLE 1

16log4loglog3 555 x

EXAMPLE 2

2)6(loglog 44 xx

EXAMPLE 3

3log27loglog2 777 x

EXAMPLE 4

125log5loglog4 222 x

EXAMPLE 5

7loglog42log 333 n

EXAMPLE 6

9loglog2 55 x

9.4 C

OMMON

LOGARIT

HMS

COMMON LOGARITHMS

Logarithms with base 10 are common logs

You do not need to write the 10 it is understood

Button on calculator for common logs

100logLOG

EXAMPLES: USE CALCULATOR TO EVALUATE EACH LOG TO FOUR DECIMAL PLACES

1. log 3 2. log 0.2

3. log 5 4. log 0.5

SOLVE LOGARITHMIC EQUATIONSExample 5:

The amount of energy E, in ergs, that an earthquake releases is related to is Richter scale magnitude M by the equation logE = 11.8 + 1.5M. The Chilean earthquake of 1960 measured 8.5 on the Richter scale. How much energy was released?

Example 6:

Find the energy released by the 2004 Sumatran earthquake, which measured 9.0 on the Richter scale and led to the tsunami.

HELPFUL HINT

If both sides of the equation cannot be easily written as powers of the same base you can solve by taking the log of each side!

EXAMPLE

3x=11 4x=15

SOLVING INEQUALITIES

Example 7

53y<8y-1

EXAMPLE 8

32x>6x+1

EXAMPLE 9

4y<52y+1

CHANGE OF BASE FORMULA

5log

12log12log

10

105

EXAMPLE

Express in terms of common logs, and then approximate its value to four decimal places.

log425 log318 log7 5

9.5 B

ASE E A

ND NAT

URAL

LOGS

NATURAL BASE EXPONENTIAL FUNCTION

An exponential function with base e e is the irrational number 2.71828…

*These are used extensively in science to model quantities that grow and decay continuously

Calculator button ex

EVALUATE TO FOUR DECIMAL PLACES

1. e2 2. e-1.3 3. e1/2

THE LOG WITH BASE E IS A NATURAL LOGWritten as : ln

y=ln x is the inverse of y = ex

All properties for logs apply the same way to natural logs

Calculator button lnx

EXAMPLES

Use calculator to evaluate to four decimal places

4. ln4 5. ln0.056. ln7

EXAMPLE

Write an equivalent exponential or log equation to the given equation.

7. ex=5 8. lnx≈0.6931

REMEMBER…..

All log properties apply to natural logs

Do the same thing for ln problems that you do for log problems

Let’s solve!!!!!!!!!

EXAMPLE 9

Solve e4x=120 and round to four decimal places

EXAMPLE 10 EXAMPLE 11

ex-2 + 4<21 ln6x > 4

EXAMPLE 12 EXAMPLE 13

ln5x+ln3x>9 2e3x + 5 =2

9.6 E

XPONENTI

AL

GROWTH

AND D

ECAY

EQUATIONS THAT DEAL WITH E

Continuously Compounded InterestA=Pert

A= amount in account after t yearst= # of yearsr= annual interest rateP= amount of principal invested

EXAMPLES

Suppose you deposit $1000 in an account paying 2.5% annual interest, compounded continuously.

Find the balance after 10 years

Find how long it will take for the balance to reach at least $1500

Suppose you deposit $5000 in an account paying 3% annual interst, compounded continuously.

Find what the balance would be after 5 years

Find how long it will take for the balance to reach at least $7000

EXPONENTIAL DECAY

y=a(1-r)t

a=initial amount r=% of decrease expressed as a decimal, this is also called

rate of decay t=time

y=ae-kt

a=initial amount k=constant t=time

EXAMPLE 3

A cup of coffee contains 130 milligrams of caffeine. If caffeine is eliminated from the body at a rate of 11% per hour, how long will it take for half of this caffeine to be eliminated?

EXAMPLE 4

The half-life of Sodium-22 is 2.6 years. What is the value of k and the equation of decay for

Sodium-22?

A geologist examining a meteorite estimates that it contains only about 10% as much Sodium-22 as it would have contained when it reached Earth’s surface. How long ago did the meteorite reach Earth?

EXPONENTIAL GROWTH

y=a(1+r)t

a= initial amount r=% of increase/growth expressed as a decimal t=time

y=aekt

a=initial amount k=constant t=time

EXAMPLE 5

Home values in Millersport increase about 4% per year. Mr. Thomas purchased his home eight years ago for $122,000. What is the value of his home now?

EXAMPLE 6

The population of a city of one million is increasing at a rate of 3% per year. If the population continues to grow at this rate, in how many years will the population have doubled?

EXAMPLE 7

Two different types of bacteria in two different cultures reproduce exponentially. The first type can be modeled by B1(t)=1200e0.1532t and the second can be modeled B2(t)=3000e0.0466t where t is the number of hours. According to these models, how many hours will it take for the amount of B1 to exceed the amount of B2?