5.1 exponential functions rules for exponents if a > 0 and b > 0, the following hold true for...

5.1 Exponential Functions

Rules For ExponentsIf a > 0 and b > 0, the following hold true for all real numbers x and y.

yxyx aa. a +=1

yxy

x

aa

a. −=2

( ) xyyx aa. =3

xxx (ab)b. a =4

x

xx

b

a

b

a. =⎟

⎠

⎞⎜⎝

⎛5

16 0 =. a

x-x

a. a

17 =

q pq

p

a. a =8

5

3

x

x

xxxxx

xxx

⋅⋅⋅⋅⋅⋅

=

53−=x

2

1

x=

If we apply the quotient rule, we get:

5

3

x

x 2−=x 2

1

x=


•For any nonzero number x:

nn

xx

1=− and n

nx

x=−

1


•Examples:

4

1

2

12

22 ==−

2733

1 33

==−

4

9

2

3

3

222

=⎟⎠

⎞⎜⎝

⎛=⎟⎠

⎞⎜⎝

⎛−

945)4(54

5

xxxx

x=== +−−

−



Examples:

•40 = 1•2-1 = ½•(½)-2 = 4•5-2 = 1/25

•(5x-2)3 = 125x-6=125/x6

•(3x/y3)2 = 9x2/y6

•(4x)-1 = 1/(4x)•(2a3b-3c4)3 = 8a9b-9c12

Simplify: 24 16= 4=

25 25= 5=

( )23− 9= 3= 3−=


( ) defined is if bbb ,2

=

bb =2


12xSimplify:

( )26x=Rewrite: 6x=Notice: 6212 =÷

2


16xSimplify:

( )28x=Rewrite: 8x=Notice: 8216 =÷

2


3 15xSimplify:

( )3 35x=Rewrite: 5x=Notice: 5315 =÷


If abn = Then ban =

3273 −=− Since( ) 273 3 −=−

Examples

2164 = Since 1624 =

11 =n Since 11 =n


In general, if n is a multiple of m, then

( )mnm n xx ÷= If n is odd

( )mnm n xx÷

= If n is even


Use the rules for exponents tosolve for x

•4x = 128•(2)2x = 27

•2x = 7•x = 7/2

•2x = 1/32•2x = 2-5

•x = -5


•(x3y2/3)1/2

•x3/2y1/3

•27x = 9-x+1

•(33)x = (32)-x+1

•33x = 3-2x+2

•3x = -2x+ 2•5x = 2•x = 2/5



Definition Exponential Function

Let a be a positive real number other than 1,the function f(x) = ax is the exponential function with base a.

5

4

3

2

1

-2

-3

-4

-5

y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

x

y = 2 xIf b > 1, then the graph of b x will:

•Rise from left to right.

•Not intersect the x-axis.

•Approach the x-axis.

•Have a y-intercept of (0, 1)


5

4

3

2

1

-2

-3

-4

-5

y

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

x

y = (1/2) xIf 0 < b < 1, then the graph of

b x will:

•Fall from left to right.

•Not intersect the x-axis.

•Approach the x-axis.

•Have a y-intercept of (0, 1)



xef(x) =

Natural Exponential Function where e is the natural base and e 2.718…

x

x x

11lim ⎟

⎠⎞

⎜⎝⎛ +=

∞→

xe


Function f(x) = 2x h(x) = (0.5)x g(x) = ex

Domain

Range

Increasing or Decreasing

Point Shared On All Graphs

(-∞, ∞) (-∞, ∞) (-∞, ∞)

(0, ∞) (0, ∞) (0, ∞)

Inc. Dec. Inc.

(0, 1)


Use translation of functions to graph the following. Determine the domain and range

f (x) = 2(x + 2) – 3 Domain (-∞, ∞)Range (-3, ∞)


Definitions Exponential Growth and Decay

The function y = k ax, k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1.

h

t

Obyy =y new amountyO original amountb baset timeh half life


An isotope of sodium, Na, has a half-life of 15 hours. A sample of this isotope has mass 2 g.

(a) Find the amount remaining after t hours.

(b) Find the amount remaining after 60 hours.

• a. y = yobt/h

• y = 2 (1/2)(t/15)

• b. y = yobt/h

• y = 2 (1/2)(60/15)

• y = 2(1/2)4

• y = .125 g


A bacteria double every three days. There are 50 bacteria initially present

(a) Find the amount after 2 weeks.

(b) When will there be 3000 bacteria?

• a. y = yobt/h

• y = 50 (2)(14/3)

• y = 1269 bacteria


A bacteria double every three days. There are 50 bacteria initially present

When will there be 3000 bacteria?

• b. y = yobt/h

• 3000 = 50 (2)(t/3)

• 60 = 2t/3

•

5.2 Simple and Compound Interest

Formulas for Simple Interest Suppose P dollars are invested at a simple interestrate r, where r is a decimal, then P is called theprincipal and P ·r is the interest received at the end of one interest period.

)1( nrPT +=


Formulas for Compound Interest After t years, the balance A in an account with principalP and annual interest rate r is given by the two formulas below.

1. For n compoundings per year:nt

n

rPA ⎟

⎠⎞

⎜⎝⎛ += 1

2. For continuous compounding: rtPeA =


Find the balance after 10 years if $1000.00 is investedat 4% and the account pays simple interest.

)1( nrPT +=

)]04(.101[1000 +=T00.1400$=T


Find the balance after 10 years if $1000.00 is investedat 4% and the interest is compounded:

a. Semiannuallynt

n

rPA ⎟

⎠⎞

⎜⎝⎛ += 1

b. Monthly:

rtPeA =c. Continuously:

nt

n

rPA ⎟

⎠⎞

⎜⎝⎛ += 1

)10*2(

2

04.11000 ⎟

⎠⎞

⎜⎝⎛ +=A

)10*12(

12

04.11000 ⎟

⎠⎞

⎜⎝⎛ +=A

$1485.95

$1490.83

$1491.82)10*04(.1000eA =

5.3 Effective Rate and Annuities

Effective Annual RateThe effective annual rate of ieff of APR

compounded k times per year is given by theequation

Another name for effective annual rate iseffective yield

11 −⎟⎠⎞

⎜⎝⎛ +=

k

eff kr

i


What is the better rate of return,7% compounded quarterly or 7.2 % compounded semianually?

11 −⎟⎠⎞

⎜⎝⎛ +=

k

eff kr

i


14

07.1

4

−⎟⎠⎞

⎜⎝⎛ +=effi

12

072.1

2

−⎟⎠⎞

⎜⎝⎛ +=effi

1.071 – 1 = .071 = 7.1%

1.073 – 1 = .073 = 7.3%

7.2% compounded semiannually is better.


What is the better rate of return,8 % compounded monthly or 8.2 % compounded quarterly?

11 −⎟⎠⎞

⎜⎝⎛ +=

k

eff kr

i


112

08.1

12

−⎟⎠⎞

⎜⎝⎛ +=effi

14

082.1

4

−⎟⎠⎞

⎜⎝⎛ +=effi

8.3%

8.5%

8.2% quarterly is better.


Future Value of an Ordinary Annuity

The Future Value S of an ordinary annuity

consisting of n equal payments of R dollars,each with an interest rate i per period is

i

iRS

n 1)1( −+=


Suppose $25.00 per month is investedat 8% compounded quarterly. How much willbe in the account after one year?

•1st quarter $25.00•2nd quarter $25.00(1+.08/4)+ $25.00 = $50.50 •3rd quarter $50.50(1+.08/4)+ $25.00 = $76.51•4th quarter $76.51(1+.08/4) + $25.00 = $103.04


Present Value of an Ordinary Annuity

The Present Value A of an ordinary

annuity consisting of n equal payments of R dollars, each with an interest rate i per period is

i

iRA

n−+−=

)1(1

5.4 Logarithmic Functions

The inverse of an exponential function is called a logarithmic function.

Definition: x = a y if and only if y = log a x


•log 4 16 = 2 ↔ 42 = 16

•log 3 81 = 4 ↔ 34 = 81

•log10 100 = 2 ↔ 102 = 100


Sketch a graph of f (x) = 2x and sketch a graph of its inverse. What is the domain and range of the inverse of f.

Domain: (0, ∞)Range: (-∞, ∞)


The function f (x) = log a x is called a logarithmic function.

•Domain: (0, ∞)•Range: (-∞, ∞)

• Asymptote: x = 0• Increasing for a > 1

• Decreasing for 0 < a < 1• Common Point: (1, 0)

Find the inverse of g(x) = 3x.

xxg 31 log)( =−


(0,1)

(1,3)

(3,1)(1,0)

(-1,1/3)

(1/3,-1)

Note: The function andit’s inverse are symmetricalabout the line y = x.

Find the inverse of g(x) = ex.

xxxg e lnlog)(1 ==−


ln x is called the natural logarithmic function

So10 =b 01log =b 01log3 =

Sobb =1 1log =bb 110log =

Soxx bb = xb xb =log 3ln 3 =e

So xb xb =logxx bb loglog =

510 5log =


1. loga(ax) = x for all x 2. alog ax = x for all x > 03. loga(xy) = logax + logay4. loga(x/y) = logax – logay5. logaxn = n logax

Common Logarithm: log 10 x = log xNatural Logarithm: log e x = ln x

All the above properties hold.



Product Rule

( ) nmnm bbb logloglog +=⋅

( )49log36log 33 ⋅=

4log24log9log 333 +=+=

Quotient Rule nm

n

mbbb logloglog −=⎟

⎠

⎞⎜⎝

⎛

225log5 ==

⎟⎠

⎞⎜⎝

⎛=−2

50log2log50log 555


Power Rule ( )mcm b

cb loglog ⋅=

ee ln4ln 4 ⋅=

414 =⋅=


Expand

⎟⎟⎠

⎞⎜⎜⎝

⎛z

yx 23

5log

zyx 52

53

5 log)log(log −+=

zyx 555 log)log2log3( −+=



maxy =maxy lnln =

mxay lnlnln +=

Find an equation of best fit for the data(1,3), (2,12), (3,27), (4,48)

xmay lnlnln +=

1lnln3ln ma +=

aln3ln =

xmy ln3lnln +=

2ln3ln12ln m+=

2ln3ln12ln m=−

m2ln4ln =2=m3=a

23xy =

5.5 Graphs of Logarithmic Functions

The function f (x) = log a x is called a logarithmic function.

•Domain: (0, ∞)•Range: (-∞, ∞)

• Asymptote: x = 0• Increasing for a > 1

• Decreasing for 0 < a < 1• Common Point: (1, 0)

The natural and common logarithms can be found on your calculator. Logarithms of other bases are not. You need the change of base formula.

a

xx

b

ba log

loglog =

where b is any other appropriate base.(usually base 10 or base e)



Sketch the graph of

)2(log3 −= xy

Domain (2,)Range (-, )

3 0

5 1

11 2

29 3


Sketch the graph of

)42(log2 += xy

Domain (-2, )Range (- , )

)2(2log2 += xy

)2(log2log 22 ++= xy

)2(log1 2 ++= xy


Sketch the graph of

)3ln(1 +−−= xy

Domain (-3, )Range (- , )


On the Richter scale, the magnitude R of an earthquake can be measured by the intensity model.

BT

aR +⎟

⎠⎞

⎜⎝⎛=log

R = Magnitudea = AmplitudeT = PeriodB = Damping Factor


BT

aR +⎟

⎠⎞

⎜⎝⎛=log

What is the magnitude on the Richter scale of anearthquake if a = 300, T = 30 and B = 1.2?

2.130

300log +⎟

⎠⎞

⎜⎝⎛=R

2.110log +=R

2.11+=R

2.2=R

Solve: 4 3x = 16 x – 2

The bases can be rewritten as: (22) 3x = (24) (x – 2)

2 6x = 2 4x – 8

6x = 4x – 8 2x = -8 x = -4

5.6 Solving Exponential Equations


To solve exponential equations, pick a convenient base (often base 10 or base e) and take the log of both sides.

Solve:

2143 =x

2143 =x

•Take the log of both sides: ( ) 21log4log 3 =x

•Power rule: ( ) 21log4log3 =x


•Solve for x:

( ) 21log4log3 =x

•Divide:

( ) 21log4log3 =⋅x

4log3

21log=x 732.0≈x


To solve logarithmic equations, write both sides of the equation as a single log with the same base, then equate the arguments of the log expressions.

Solve:

( ) ( ) ( )2log1log82log +=+−+ xxx


•Write the left side as a single logarithm:

( ) ( ) ( )2log1log82log +=+−+ xxx

( )2log1

82log +=⎟

⎠

⎞⎜⎝

⎛+

+x

x

x


•Equate the arguments:

( )2log1

82log +=⎟

⎠

⎞⎜⎝

⎛+

+x

x

x

21

82+=⎟

⎠

⎞⎜⎝

⎛++

xxx


•Solve for x:2

1

82+=⎟

⎠

⎞⎜⎝

⎛++

xxx

( ) ( )( )121

821 ++=⎟

⎠

⎞⎜⎝

⎛+

++ xx

x

xx


( )( )1282 ++=+ xxx

( )( )230 −+= xx

2382 2 ++=+ xxx

60 2 −+= xx


( )( )230 −+= xx 2 ,3−=x

( ) ( ) ( )2log1log82log +=+−+ xxx

•Check for extraneous solutions.

•x = -3, since the argument of a log cannot be negative


To solve logarithmic equations with one side of the equation equal to a constant, change the equation to an exponential equation

Solve:

64loglog 2 =+ xx


•Write the left side as a single logarithm:

64loglog 2 =+ xx

( ) 64log 2 =⋅ xx

( ) 64log 3 =x


( ) 64log 3 =x

•Write as an exponential equations:

63 104 =x


•Solve for x:

2500003 =x

99.62=x


63 104 =x

5.1 exponential functions rules for exponents if a > 0 and b > 0, the following hold true for...

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