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Chapter 8 – Exponential and Logarithmic Functions
18 Days
8.1 Modeling Exponential Functions
Five Days
What equations did you come up with to model your data in the penny lab?
Penny Lab
Exponential Functions
decay. lexponentia have we1,b0 If
growth lexponentia have we1,b If
.100 , where
form theoffunction a isFunction lExponentiaAn
, b, bax
bay x
Family of Exponential Functions
Identify the following as either exponential growth, decay, or undefined.
Growth or Decay
x
x
x
x
y
y
y
y
)4(3
)10(5.
)25(.1300
2
Graphing Exponential Functions
x
x
y
y
)4(5.
)25(.100
Lower Dauphin HS had a student population of 1150 kids in the school year ending in 2010. Find the function that models the population if the rate of growth continues at a rate of 2.3% per year.
Modeling Exponential Growth
Find the equation of the exponential function that passes through (4,8) and (6,32).
Writing equations given two points.
Questions??
pg 434 (# 1 - 19 odd)
Homework
pg 434 (32, 42 - 45 all)
Homework
Practice with Exponential Growth and Decay
Homework
8.2 Properties of Exponential Functions
Four Days
Exponential functions are those that have
An example of an exponential function is
What is an exponential function?
power variablebaseconstant
ky
),(
),(
:Asymptote Horizontal
0 :Range
:Domain 14
12
10
8
6
4
2
-2
-15 -10 -5 5 10 15
f x = 2x
kbaf(x) x-h
xxf 2)(
Asymptote Horizontal theis
point.locator theis )1,( where)(
ky
kahkbaxf hx
We can shift exponential function using the same patterns from before.
Shifting Exponential Graphs
Asymptote Horizontal theis
point.locator theis )1,( where)(
ky
kahkbaxf hx
xxf 24)(
224)( xxf
324)( 2 xxf
The half life of a radioactive substance in the time it takes for half of the substance to decay.
Half Life
units) life half as same thebemust (units Time
substance of life Half
substance ofAmount Initial 0
21
0
1
x
h
A
Ay xh
Caffeine has a half life of 5.7 hours in the human body. If you drink a Coca-Cola at noon, how much caffeine will be in your body when you go to bed?
What do we need to know?
What time do you go to bed?
32mg of caffeine per 12oz can.
Half Life Example
The exponential function with base e are very useful in describing continuous growth and decay.
The Natural Function; e
71828.2
1lim 1
e
e x
xx
The Natural Exponential Function
ns.applicatio and smathematic advancedin
functions usefulmost theof one isfunction lexponentia natural The
number x. realevery for
)(
by definded is Function lExponentia Natural Thexexf
f
What is the difference between simple interest and compound interest?
What is continuously compounded interest?
Continuously Compounded Interest
years. ofnumber theis
decimal) a (as rateinterest annual theis
invested principal theis
years after amount total theis
: where
:FormulaInterest Compoundly Continuous The
t
r
P
tA
PeA rt
An initial investment of $35000 is continuously compounded at 8.5% interest. How much is the investment worth after 5 years? After 15 years?
Continuously Compounded Interest
pg 442 (# 2-8 even, 16, 17, 24-26, 36, 37)
Homework
Exponential Growth and Decay WS
Homework
Simple and Compound Interest WS
Homework
8.1 - 8.2 Review
Homework
8.2 Solving Basic Exponential Equations
One Day
We know that in exponential functions the exponent is a variable. When we wish to solve for that variable we have two approaches we can take.
One approach is to use a logarithm. We will learn about these in a later lesson.
The second is to make use of a property called the Equality Property for Exponential Functions.
The Equality Property of Exponential Functions
Basically, this states that if the bases are the same, then we can simply set the exponents equal.
This property is quite useful when we are trying to solve equations involving exponential functions.
Let’s try a few examples to see how it works.
The Equality Property for Exponential Functions
21 xifonly and if Then
.1 and 0 that Suppose21 xbb
bbxx
32x 5 3x 3(Since the bases are the same wesimply set the exponents equal.)
2x 5 x 3x 5 3
x 8
Here is another example for you to try:
Example 1a:
23x 1 21
3x 5
Example 1
How can we solve an equation when the bases are not the same??
32x 3 27x 1
Does anyone have an idea howwe might approach this?
32x 3 27x 1
32x 3 33(x 1) (our bases are now the sameso simply set the exponents equal)2x 3 3(x 1)
2x 3 3x 3
x 3 3
x 6
x 6
Let’s try another one of these.
Example 2
32
116 1 x
Example 3
Example 4122 84 xx
Example 51331 2733 xxx
Solving Exponential Equations WS
Homework
8.3 LogarithmsTwo Days
The logarithm to the base b exponential is defined as:
What is a Logarithm?
xyby bx log
Re-writing in logarithmic form
32=9
xa+b=9
364log4
rGM log
Evaluate
Evaluating Logsx16log8
Evaluate
Evaluating Logsx27log9
The common logarithm is a log that uses base 10. It can be written in either of the two forms:
The Common Logarithm
yy logor log10
Evaluate the following logs:
Evaluating Logs with the Calculator
3
4log
45log
20log
10log
Page 450 (#6-25 all, 41-48 all, 53-61 all)
Homework
8-3 WS (# 1 - 49 odd)
Homework
8.4 Properties of Logarithms
One Day
Properties of Logarithms
Write each logarithmic expression as a single logarithm
zyx
zyx
x
logloglog3
logloglog3
2loglog
2log5log
Expand each logarithm
42
2
2
3
25
log
log
log
2log
zy
wx
z
xy
yx
x
We can use the properties of logs to re-write a logarithmic expression as a single log so that we can evaluate, or solve, the logarithm.
Ex:
Why use properties of logs?
5log21log 5521
Evaluate
Your turn…
4log2log5 22
Questions??
pg 457 (# 11 - 30 all, 33-41 odd)
Homework
8.5 Solving using the Equality Property
Four Days
Property of Equality for Logs
nmnm bb loglog
Solving Logs
xx log2log27log
Solving Logs
8-4 WS (# 14 - 40 even); 7-5 WS (# 11 - 21 odd)
Homework
8.5 Solving Exponential and Logarithmic Equations
Four Days
Guidelines:◦ When solving an exponential equation, you must
isolate the exponential (or write as one exponential on each side) before you take the log of both sides.
◦ When solving a logarithmic equation, you must isolate the logarithm (or write as a single log on each side of the equation) before you raise both sides using a base b.
Solving Logarithmic and Exponential Equations
Logs as Inverse Operations
xxb
xxb
x
xb
b
xlnlog
x
e
and
eln log
:isThat another. one of inverses are and log xb b
Our calculators can only handle logs that are either base 10, or base e. So, we need a way to re-write any log so that it can be entered into the calculator.
The Change of Base Formula:
Change of Base Formula
b
x
b
xxb log
logor
ln
lnlog
Solving Exponentials
32 x
Solving Exponentials
14425 12 x
Solving Exponentials
1062 x
Solving Logarithms
12log x
Solving Logarithms
213log x
Solving Logarithms
94.2log6loglog3 x
Solving Exponentials231 25 xx
Pg 464 #2-44 even, SKIP 20
Homework
Logs that result in Quadratics
Pg 466 #79-96 all, 100-103 all
Homework
Exponential and Logs Review
Homework
8.6 Natural LogarithmsFour Days
7-8 WS Applications of Exp and Log Functions
Homework
7-8 WS Applications of Exp and Log Functions
Homework