Unit 8: ExPonential! Logs
Carli Castellano, Caroline Chivily, Julia Ashley
Chapter 7 Exponential and logarithmic Functjons
Section 7.1: Exploring Exponential Models
Definitions
• An exponential function is a function with the general form y= abx,a=O with b>O, and b
does not = 1. In an exponential function, the base b is a constant. The exponent x is the
independent variable with domain the set of real numbers.
• For exponential growth,as the value x increases, the value of y increases
• For exponential decaY,as the value of x increases, the value of y decreases, approaching
zero.
• An asymptote is a line that a graph approaches as x or y increases in absolute value
• For exponential growth y=ab x, with b>1 the value b is the growth factor
• For exponential decay 0<b<1 and b is the decay factor.
Key Concepts
• if a>O and b>1 the function represents exponential growth
• if a> 0 and 0<b<1 the function represents exponential decay
• in either case the y-intercept is (O,a) the domain is all real numbers, the asymptote is
y=O, and the range is y>O
• You can model exponential growth or decay with this function- A(t)=a(1+r)t
• For growth or decay to be exponential,a quantity changes by a fixed percentage each
time period
Example 1: Graphing an Exponential function
What is the graph of y=2x
Step 1: Make a table of values
x
2x
Y
-4
2""
1/16=0.0625
0
2°
1
-3
2-3
1/8= 0.125
1
21
2
-2
2-2
1/4= 0.25
2
22
4
-1
2- 1
1/2= 0.5
3
23
8
Example 2: Identifying Exponential Growth and Decay
Without graphing determine whether the function represents exponential growth or decay. Then find the y~intercept. f(x)= 2(O.65X) -exponential decay - Y-int: (0,2) y=3(4X) - exponential growth - (0,11) Example 3: Modeling Exponential Growth
Suppose you invest $500 in a savings account that pays 3.5% annual interest. How much will be
in the account after five years?
a=500
b=(1 +0.035)= 1.035
y=500(1.035)t _ y=500(1.035)5 = $593.84
Example 5: Writing an Exponential Function
The table shows the world population of Iberian lynx in 2003 and 2004. If the trend continues and
the population is decreasing exponentially, how many Iberian lynx will there be in 2020.
Year
Population
2003
150
2004
120
y=abX=a( 1+r)X
1) Define the variables
x= number of years since 2003
y=population of lynx
2) Solve for a and b. Use y=abx
150=abo - a= 150
120= ab 1 _ 120=150b - b=0.8
3) Find the value of r.
1+r=b _1+ r= 0.8 - r= -0.2
4) Write the model
y=150(0.8)X
5) Use the model to find the world lynx population in 2020.
x=2020-2003=17
y=150(0.8)17
y= 3.37 - 3 lynx will be left
Section 7.2: Properties of Exponential Functions
Essential Understanding: The factor a in y=ab x can stretch or compress and possibly reflect the
graph of the parent function y=bx
Concept Summary:
• Parent function
• stretch ({a} >1) compression (shrink) (0< {a} <1) Reflection (a <0) in x-axis • Translations (horizontal by h; vertical by k) • All transformations combined
y=b(X-h) +k
y= ab(x-h) +k
Definitions: Natural base exponential: are exponential functions with base e. These functions are useful for describing continuous growth or decay. Exponential functions with base e have the same properties as other exponential functions. Example 1: Graphing y=aJY How does the graph of y= -1/3 times 3x compare to the graph of the parent function? Step 1: Make a table of values x
y=3 x
y=-1/3(3 X)
-2
1/9
-1/27
-1
113
-1/9
0
1
-1/3
1
3
-1
2
9
-3
The - 113 in y= -1/3(JX) reflects the graph of the parent function y=3x across the x-axis and
compresses it by the factor 1/3. The domain and asymptote remain unchanged. The y-intercept
becomes -1/3 and the range becomes -1/3 and the range becomes y<O.
Example 2: Translating the Parent Function y=b x
How does the graph of each function compare to the graph of the parent function?
y= 2(x-4)
Step 1: Make a table of values
x
y=2x
x
y=2x
-2
1/4
1
2
-1
1/2
2
4
0
1
3
8
The (x-4) in y=2(x-4) translates the graph of y=2x to the right 4 units. The asymptote remains y=O.
The y-intercept becomes 1/16.
Example 3: Using an Exponential Model
Time (Min.)
Temp.(OF)
Temp. - Room Temp. (OF)
0
203
135
5
177
109
10
153
85
15
137
69
20
121
53
25
111
43
30
104
36
(Find an exponential function to model the data. Use the list feature on the graphing calculator.
Assume that room temperature is 68°)
y=134.5(0.956Y + 68
The initial temperature of a coffee cup of coffee is 203°F. An exponential model for the
temperature y of the coffee after x minutes is y=134.5(0.956Y + 68. How long does it take for the
coffee to reach a temperature of 100 OF?
(Enter your function into your calculator and go to your table and locate 100 (or closest number
to 100) in the y column. (Go to tableset and and .4Tbl if necessary))
IN 39.1 MINUTES THE COFFEE WILL BE COOLED TO 1000 F
Section 7.3: Logarithmic Functions as Inverses
• Recall: x=b Y logbx= y •
•
The inverse of an exponential function is called a logarithmic function.
For a logarithm, y is the exponent that you need to raise b to get x.
Example 1: Writing Exponential Equations in Logarithmic Form
1} 100=102
log10100=2
2} 81=34
log381=4
3) 36=62
log636 =2
Example 2:Evaluating a Logarithm
1) log832
8 Y=32
2 3Y=2 5
3y=5
y=5/3
2) log5125
5Y=125
5Y=53
y=3
3} log432
4 Y=32
2 2Y=2 5
2y=5
y=5/2
Example 3: Using a Logarithmic Scale
In December 2004, an earthquake with magnitude 9.3 on the Richter scale hit off the northwest
coast of Sumatra. The diagram shows the magnitude of an earthquake that hit Sumatra in March
2005. The formula log 1/12=M1-M2 compares the intensity levels of earthquakes where I is the
intensity level determined by a seismograph, and M is the magnitude on a Richter scale. How
many times more intense was the December
earthquake than the March earthquake?
log 1/12=M1-M2 (Use the formula.) log 1/12=9.3-8.7 (Substitue M1=9.3 and M2=8.7.) log 1/12=0.6 (Simplify) 1/12=10°.6 (Apply the definition of common logarithm.) =4 (Use your calculator.) A logarithmic function is the inverse of an exponential function.
Recall: We can graph the inverse of a function by switching the x and y coordinates.
Example 4: Graphing a Logarithmic Function
What is the graph of y=log 4x? Describe the domain, range, y-intercept and asymptotes.
4x=y is the inverse
Log: Domain: x>O
Range: all reals
y-intercept: none
VA: none
Example 5: Translating y=logtf
How does the graph of the function compare to the parent function?
1) y=log2(X-3)+4
right 3; up 4,
Domain: x>3
Range: All reals
2) y=510g 2x
stretch by 5
Domain: x>O
Range: All reals
3)y=log2(x+4)
left 4
Domain:x>-4
Range: All reals
Section 7.4: Properties of logarithms
am X an = am+n (product property)
am/an = am-n (quotient property)
(amt = amn (power property)
Let x=109bm and y=109bn
m=bX
mn=bx • bY
mn=bx+y
109bmn=109bm+109bn
Product Property: 109bmn=109bm+109bn
Quotient Propery: 109bm/n=109bm-109bn
Power Property: 109bmn=n 109bm
Example 1: Simplifying Logarithms
Make each a sin91e 109arthim.
1) 10945x+10943x
109415x2 (With product power multiply 5x and 3x to get 15x2)
2 109415x (move the 2 to the front with power property)
2) 2 10946-10949
109462-10949 (power property moves the 2 to the front)
109436-10949 (simplify 62 to 36)
109436/9 (quotient property to divide) 10944 (simplify) 4 Y=4 (solve for y) y=1
3) 109432-10942
109432/2 (quotient property of 109arithms)
109416 (divide)
109442 (write 16 as a power of 4)
2 (simplify)
Example 2: Expanding Logarithms
1) 1093250/37
log3250-log337(quotient property) log32(125)-log337 (product property) log32+log3125-log337 (product property) log32+log353-log337 (power property) log32+310g35-log337 (power property) 2) log39x5
IOg39+log3X5 (product property)
log39+510g3x (power property)
Change of base: 10gbm=logm/logb
Example 3: Using the Change of Base Formula
1) log632
log32/8
1.6
2) log418
log18/4
2.085
Example 4: Using a Logarithmic Scale
The pH of a substance equals -Iog[H+]. where [H-] is the concentration of hydrogen ions.
Suppose the hydrogen ion concentration for Substance A is twice that for Substance B. Which
substance has a greater pH level? What is the greater pH level minus the lesser pH level?
Explain.
Substance A: 2-[H+ e]
Substance B: [H+e]
A: pH=-log[2([H+ e])]
B: pH=-log[H+e]
pH of A =-(Iog 2+ log [H+eD
pH of A =-log2-log[H+B]
pH of A =-log2+pH of B
pH of A =pH of B-Iog 2
pH of B is bigger
Section 7.5: Exponential and Logarithmic Equations
Example 1: Solving an Exponential Equations- Common Base
1) 273X= 81
3 3*3X=34
=
39x 34
9x=4 (take the exponents)
x=4/9
=
2) 5 3X 1/125
53X= 5-3
(53 = 125; 5-3 = 11125)
3x=-3
x= -1
• When bases are not the same, you can solve an exponential equation by taking the
logarithm of each side
Example 2: Solve an Exponential Equation
1) 5 2x= 130
log52x= log 130 (Make a log with a common base of 10) 2xlog5= log 130 (Power Property moves the 2x) 2x= log130/10g5 (Isolate the 2x by dividing log5 on both sides) x= (log1301l0g5)/2 (Isolate x by dividing by 2 on both sides) x= 1.512 2) 8 + 1OX= 1008 10x= 1000 (Isolate 10x by subtracting 8 from both sides) log1OX= log1000 (Make a log with a common base of 10) xlog10= log1000 (Power Property moves the x) x= log10001l0g10 (Isolate x by dividing log10 on both sides) x=3 Example 3: Solving an Exponential Equation with a Graph or Table
1) 4 7x= 250
log447x= log4250 (Put log4 to cancel out the 4 and isolate the 7x)
7x= log250/10g4 (Change of base) x= (log2501l0g4)/7 (Divide 7 on both sides to isolate the x) x= 0.56898 Graphically: y= 47X y=250 (Find the pOint of Intersection using your graphing calculator) Point of Intersection: (0.56898, 250)
2) 6 x= 4565
log66x= log64565 (Put log6 to cancel out the 6 and isolate the x) x= log4565/1og6 (Change of base) x= 4.7027 Graphically: y=6 x y=4565 (Find the point of intersection using your graphing calculator) Point of Intersection: (4.7027, 4565) Example 4: Modeling with an Exponential Equation
Wood is a sustainable, renewable, natural resource when you manage forests properly. Your lumber company has 1,200,000 trees. You plan to harvest 7% of the trees each year. how many years will it take to harvest half of the trees? T(n)= a(b)n T(n)=600,000 a=1,200,000 b= 1+ -.0.07=0.93 T(n)=1,200,000(0.93t 600.000=1,200,000(0.93)n 0.5=(0.93)" (Divide both sides by 1,200,000 to isolate term with n) logo.930.5= logo.93(O.93)n (Put logo.93 to cancel out 0.93 and isolate n) logO.5/logO.93= n n= 9.55 Example 5: Solving
REMEMBER: bY=x
a Logarithmic Equation
1) log2x=-1
10-1= 2x (Put into exponential form)
10-1/2 = x (Divide by 2 on both sides to isolate x)
x= 0.05
2) log(3x+1)= 2
102 = 3x+1 (Put into exponential form)
99= 3x (Subtract 1 from both sides to isolate the 3x)
x= 33 (Divide by 3 on both sides to isolate the x)
Example 6: Using Logarithmic Properties
1) logx- log3= 8
logxl3= 8 (Quotient Property)
10 8= xl3 (Put in Exponential Form)
x= 300000000 (Multiply 3 on both sides)
2) log2x+ logx= 11 log2x(x)= 11 (Product Property) log2x2= 11 (Simplify) 1011 = 2X2 (Put into Exponential Form) 1011 /2= x2 (Divide 2 on both sides) x2= 50000000000 x= 223606.8 (Square Root) Section 7.6: Natural Logarithms
•
The inverse or logarithmic of the exponential function y=ex is 10geY=x -+ logex=y
•
•
Natural logarithmic function is y=lnx
Use the same properties as logarithms
Example 1: Simplifying a Natural Logarithmic Expression
1) 31n 5
In 53 (Power Property)
In125
2) In 9 + In 2
In 9(2) (Product Property)
In 18
Example 2: Solving a Natural Logarithmic Equation
1) In x= 2
e 2 = x (Exponential Form)
x= e 2
Check Answer:
In e 2 = 2
2=2
2) In 2x + In 3 = 2
In 2x(3) = 2 (Product Property)
In 6x= 2
e2 = 6x (Put in Exponential Form)
x= e 2/6 (Divide 6 on both sides to isolate x)
Check Answer:
In 2(e2/6) + In 3= 2
1m 6e2/6 = 2
2=2
Example 3: Solving an Exponential Equation
1) eX =18
In eX = In 18 (In cancels out the e and isolates the x)
x=ln18
2) e 2x= 12 In e2x = In 12 (In cancels out the e and isolates the 2x) 2x = In 12 (Divide both sides by 2 to isolate the x) x= (In 12)/2
Example 4: Using Natural Logarithms
A spacecraft can attain a stable orbit 300 km above Earth if it reaches a velocity of 7.7 km/s. The
formula for a rocket's maximum velocity v in kilometers per second is v= -0.0098l + c In R. The
booster rocket fires for l seconds and the velocity of the exhaust is c km/s. The ratio of the mass
of the rocket filled with fuel to its mass without fuel is R. Suppose the rocket shown in the photo
has a mass ratio of 25, a firing time of 100 s and an exhaust velocity as shown. Can the
spacecraft attain a stable orbit 300 km above Earth?
Let R= 25, c= 2.8, and t= 100. Find v.
v=
v=
v=
v=
-0.0098l + c In R (Use the formula) -0.0098(100) + 2.8 In 25 (Substitute) -0.0098 + 2.8(3.219) 8.0 Spacecraft can attain a stable orbit
Jeopardy! !!!! Exponential
Growth
and Decay
Vocabl
Vocabl
Vocabl
Transfonnations
A(t)=P(e)rt
Exponential
Form to
Logarithmic
Form
pH scale!
(pH= -log[H1
Properties of
Logarithms
Solving
Exponential
Equations
of Different
and
Common
Bases
Sollling
Logarithmic
and Natural
Logarithmic
Equations
Simplifying
Natural
Logarithmic
Expressions
$200
Determine
whether it is
growth or
decay.
y=12(O.95)X
Define:
Exponential
growth.
y=2(X-4)
p= $131
r-7%
increase
t= 10 years
Find A(t)
103=1000
What is the pH
of a solution
with aW
concentration
of 1.5 x 10-3
M?
What
property is
used
101432-10942
How do you
use your
calculator
doing this
problem
log81 27?
Use your
calculator to
evaluate e3
Inx=2
$400
Determine
whether it is
growth or
decay.
y=0.25(2)X
Define:
Change of
Base
Formula.
y=3(x+5)
p= $5000
r- 10%
increase
t= 35 years
Find A(t)
625= 54
What is W
concentration
of a solution
with a pH of
4.61?
What property
is used
109,4- log,16?
How do you
use your
calculator
doing this
problem
log536?
What is the
value of
log832?
4e2 x+2=16
$600
Determine
whether it is
growth or
decay and
state the
y-intercept.
y=3(4X)
Define:
Logarithm.
y=-1/3(3X)
Suppose you
won a contest
at the start of
5th grade that
deposited
$3000 in an
account that
pays 5%
annual interest
compounded
continuously.
How much will
you hallein
the account
82= 64
What is the pH
of a solution
with anW
concentration
of2x10-2 M?
What are the
properties
used log.
x""29?
Use the
change of
base formula
to evaluate
the
expression
log3 33
What is the
value of
log42?
In(x-3)2=4
when you
enter high
school 4 years
later?
$800
Suppose you
invest $500 in
a savings
account that
pays 3.5%
annual
interest. How
much will be in
the account
after 5 years?
Define:
Natural
Logarithmic
Function.
y=3(2X)
A student
wants to save
$8000 for
college in five
years. How
much should
be put into an
account that
pays 5.2%
annual interest
compounded
continuously?
49= 72
What is the W
concentration
of a solution
with a pH of
6.30?
What are the
properties
used log
4xfy?
How do you
set up
IOg8127 using
a common
base?
What is the
value of log4
87
21n15-ln75
$1000
Suppose
you invest
$500 in a
savings
account that
pays 3.5%
annual
interest.
When will
the account
contain at
least $6507
Define:
Exponential
Equation.
y=5(0.25")+5
How long
would it take to
double your
principal in an
account that
pays 6.5%
annual interest
compounded?
10-2= 0.01
What is the
pOH ofa
solution with
anW
concentration
of 3.7 X 10-7
M?
(Hint: Find the
pH and
subtract from
14 to find
pOH)
What are
the
properties
used 610g 2
x+ 510g 2 y?
What is the
answer for
log81 27?
What is the
value of
log5 125
In2x+ln3=2
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