Transcript Ch 5 Exponential and Logarithmic Functions

5-3
LOGARITHMIC
FUNCTIONS
Logarithms have many applications inside and
outside mathematics. Some of these occurrences
are related to the notion of scale invariance. For
example, each chamber of the shell of a nautilus is
an approximate copy of the next one, scaled by a
constant factor
 •To find the number of payments on a loan or the time to reach an investment
goal
 •To model many natural processes, particularly in living systems. We perceive
loudness of sound as the logarithm of the actual sound intensity, and dB
(decibels) are a logarithmic scale. We also perceive brightness of light as the
logarithm of the actual light energy, and star magnitudes are measured on a
logarithmic scale.
 •To measure the pH or acidity of a chemical solution. The pH is the negative
logarithm of the concentration of free hydrogen ions.
 •To measure earthquake intensity on the Richter scale.
 •To analyze exponential processes. Because the log function is the inverse of the
exponential function, we often analyze an exponential curve by means of
logarithms. Plotting a set of measured points on “log-log” or “semi-log” paper
can reveal such relationships easily. Applications include cooling of a dead body,
growth of bacteria, and decay of a radioactive isotopes.The spread of an epidemic
in a population often follows a modified logarithmic curve called a “logistic”.
 •To solve some forms of area problems in calculus. (The area under the curve
1/x, between x=1 and x=A, equals ln A.)

Other uses that are more aligned with the scope of
this course are determining decibel levels for
sound (loudness) and severity of earthquakes.
Logarithms are used in other areas but the math
needed to apply them is advanced beyond this class.
We will use a few of these concepts to learn how
to deal with logs through that use hopefully you
learn an appreciation for the need of logs.
First of all, what is a logarithm? You’ve worked with them before, you’ve learned the rules
before, but do you really know what one is?
 In its simplest form, a logarithm answers the question:
 How many of one number do we multiply to get
another number?
For example how many 2s do you need to multiply together
to get 8?
The log then would be 3.
The notation would be
log2 8  3
How many of the bases “2” do you
need to multiply by itself to get 8?
REMEMBER LOGS AND EXPONENTIALS
ARE INVERSES OF ONE ANOTHER.
 Take your calculator, find the log button, the log button on your calculator
uses a base of 10. Later we can change that, but for right now our calculator
is only useful when the base is 10.
log10 6.3  0.8
Verify the above value with your calculator.
What does this mean.
 To get a value of 6.3, you need to multiply 10 by itself 0.8 times.
COMMON LOG
 A log that involves a base of 10 is referred to as the
common log. The definition is as follows ….
log x  a iff 10  x
a
GENERAL LOG RULE FOR LOGS OF BASES
OTHER THAN 10
logb x  a iff x  b
a
log10 1000  3
log 2 32  5
log10 0.1  1
1
log16 4 
2
WRITE THE FOLLOWING IN LOG FORM
103 = 1,000 42 = 16 33 = 27 51 = 5 70 = 1 4-2 = 1/16 251/2 = 5
x  b iff logb x  a
a
PROPERTIES OF LOGS
log a 1  0
log a a  1
log a a  x
x
a
log a x
x
HWK. PG. 194
2-8 EVEN
11-16
28A,
LAWS OF LOGS
log a ( AB)  log a A  log a B
 A
log a    log a A  log a B
B
log a ( A )  C log a A
c
WRITE IN EXPANDED FORM
log 2 (6 x)
log 5
3
6
log 5 ( x y )
WRITE IN CONDENSED FORM AS ONE
LOGARITHM
1
3log x  log( x  1)
2
WRITE AS ONE LOGARITHM
1
2
4 log x  log( x  1)  2 log( x  1)
3
USE THE LAWS OF LOGS TO EVALUATE
EXPRESSIONS
log4 2  log4 32
 log4 (2  32)
 log4 64

NATURAL LOG “LN”
We can take a log that has a base of e. It would look like
the following
loge x
However this is used so often that it gets its own notation
loge x  ln x
This is read as the natural log of x.
WE USE THE SAME LAWS FOR NATURAL
LOG. AS WELL AS THE SAME PROPERTIES
ln1  0
ln e  1
ln e  x
x
e
ln x
x
HWK
SOLVING LOGARITHMIC EQUATIONS
 In order to solve logarithmic equations the bases must be
consistent throughout the entire equation.
 If the equation has parts that needs to be condensed
through the use of the laws of logarithms you must do that
first. Ultimately we want one of two situations to arise,
either have 0 or 1 logs on each side of the equation with
the same base.
EXAMPLE
log 2 x  log 2 5
 This is very generic, but if both logs have the same bases then we can
ignore the log part and set x=5. Thus arriving at our solution.
 Obviously x and 5 can be more complex expressions.
log4 x  log4 (2 x 1)
2
 We can also have equations where we must first condense one or
both sides of the equation.
2logb x  logb (4)  logb ( x 1)
ln( x  1)  ln( x  2)  1
 Need to re-write so that e shows up
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Slide 5.3-26
Chapter 5: Exponential and Logarithmic
Functions
5.1
5.2
5.3
5.4
5.5
Inverse Functions
Exponential Functions
Logarithms and Their Properties
Logarithmic Functions
Exponential and Logarithmic Equations and
Inequalities
5.6 Further Applications and Modeling with
Exponential and Logarithmic Functions
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Slide 5.3-27
5.3 Logarithms and Their Properties
Logarithm
For all positive numbers a, where a  1,
ay  x
is equivalent to
y  loga x.
A logarithm is an exponent, and loga x is the
exponent to which a must be raised in order to
obtain x. The number a is called the base of the
logarithm, and x is called the argument of the
expression loga x. The value of x must always be
positive.
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Slide 5.3-28
5.3 Examples of Logarithms
Exponential Form
Logarithmic Form
23  8
1 4
 2   16
51  5
3 0
4   1
Example
Solution
log 2 8  3
log 16  4
log 5 5  1
log 1  0
1
3
4
Solve log 4 x  32 .
log 4 x  32
x4
x8
3
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2
2
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5.3 Solving Logarithmic Equations
Solve a) x  log8 4 b) log x 16  4.
Example
Solution
a) x  log8 4
8x  4
2 
3 x
 22
2 2
3x  2
2
x
3
3x
b) log x 16  4
4
x  16
x   4 16
x  2
2
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Since the base must
be positive, x = 2.
Slide 5.3-30
5.3 The Common Logarithm – Base 10
For all positive numbers x,
log x  log10 x.
Example
Evaluate a) log12 b) log 0.1 c) log 53 .
Solution
Use a calculator.
a) log 12  1.079181246
b) log 0.1  1
3
c) log  .2218487496
5
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Slide 5.3-31
5.3 Application of the Common
Logarithm
Example In chemistry, the pH of a solution is defined as
pH   log[H3O ] where [H3O+] is the hydronium ion
concentration in moles per liter. The pH value is a measure of
acidity or alkalinity of a solution. Pure water has a pH of 7.0,
substances with pH values greater than 7.0 are alkaline, and
substances with pH values less than 7.0 are acidic.
a) Find the pH of a solution with [H3O+] = 2.5×10-4.
b) Find the hydronium ion concentration of a solution with
pH = 7.1.
Solution
a) pH = –log [H3O+] = –log [2.5×10-4]  3.6
b) 7.1 = –log [H3O+]  –7.1 = log [H3O+]
 [H3O+] = 10-7.1  7.9 ×10-8
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Slide 5.3-32
5.3 The Natural Logarithm – Base e
For all positive numbers x,
ln x  log e x.
•
On the calculator, the natural logarithm key is
usually found in conjunction with the e x key.
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Slide 5.3-33
5.3 Evaluating Natural Logarithms
Example
Evaluate each expression.
(a) ln12
(b) ln e10
Solution
(a) ln 12  2.48490665
(b) ln e10  10
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Slide 5.3-34
5.3 Using Natural Logarithms to Solve
a Continuous Compounding Problem
Example Suppose that $1000 is invested at 3%
annual interest compounded continuously. How
long will it take for the amount to grow to $1500?
Analytic Solution
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A  Pert
1500 1000e0.03t
1.5  e0.03t
ln1.5  0.03t
ln1.5
t
13.5 years
0.03
Slide 5.3-35
5.3 Using Natural Logarithms to Solve
a Continuous Compounding Problem
Graphing Calculator Solution
Let Y1 = 1000e0.03t and Y2 = 1500.
The table shows that when time (X) is 13.5 years, the
amount (Y1) is 1499.3  1500.
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Slide 5.3-36
5.3 Properties of Logarithms
For a > 0, a  1, and any real number k,
1. loga 1 = 0,
2. loga ak = k,
3. aloga k  k.
Property 1 is true because a0 = 1 for any value of a.
Property 2 is true since in exponential form: a k  a k .
Property 3 is true since logak is the exponent to
which a must be raised in order to obtain k.
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Slide 5.3-37
5.3 Additional Properties of Logarithms
For x > 0, y > 0, a > 0, a  1, and any real number r,
Product Rule
log a xy  log a x  log a y.
Quotient Rule
log a xy  log a x  log a y.
Power Rule
loga xr  r loga x.
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Slide 5.3-38
5.3 Additional Properties of Logarithms
Examples Assume all variables are positive.
Rewrite each expression using the properties of
logarithms.
1. log 8 x  log 8  log x
2.
15
log9  log9 15  log9 7
7
3.
1
log5 8  log5 8 2  log5 8
2
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1
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5.3 Example Using Logarithm Properties
Example Assume all variables are positive. Use the
properties of logarithms to rewrite the expression
log
3 5
x
y
n
b
zm
.
x y
x y 

log
b
m
m 
z
 z 
3 5
Solution logb n
3 5
n
3 5
x
y
1
 logb m
n
z
 1 logb x 3  logb y 5 logb z m
n
 1 3 logb x 5 logb y  m logb z 
n
 3 logb x  5 logb y  m logb z
n
n
n

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1

Slide 5.3-40
5.3 Example Using Logarithm Properties
Example Use the properties of logarithms to write
2
3
1
log
m

log
2
n

log
m
n as a single logarithm
b
b
b
2
2
with coefficient 1.
Solution
1
2
logb m  32 logb 2n  logb m2n
 log b m  log b 2n   log b m n
1
2
2


m
2
n
 log b
m2n
3
1
2
2
2
n
 log b
3
m 2
1
3
2


2
n
 log b  3   log b 8n3
m 
m
1
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3
2
2
2
3
Slide 5.3-41
5.3 The Change-of-Base Rule
Change-of-Base Rule
For any positive real numbers x, a, and b, where
a  1 and b  1,
log b x
log a x 
.
log b a
Proof Let
y  log a x.
ay  x
log b a y  log b x
y log b a  log b x
log
y  log
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x
ba
b
log
 log a x  log
x
ba
b
Slide 5.3-42
5.3 Using the Change-of-Base Rule
Example Evaluate each expression and round to
four decimal places.
(a) log5 17
(b) log 2 0.1
Solution Note in the figures below that using
either natural or common logarithms produce the
same results.
(a)
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(b)
Slide 5.3-43
5.3 Modeling the Diversity of Species
Example One measure of the diversity of species
in an ecological community is the index of diversity
H   P1 log2 P1  P2 log2 P2    Pn log2 Pn 
where P1, P2, . . . , Pn are the proportions of a sample
belonging to each of n species found in the sample.
Find the index of diversity in a community with two
species, one with 90 members and the other with 10.
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Slide 5.3-44
5.3
Modeling the Diversity of Species
Solution Since there are a total of 100 members in
the community, P1 = 90/100 = 0.9, and
P2 = 10/100 = 0.1.
log2 .9  ln .9  .152 and log2 .1  ln .1  3.32
ln 2
ln 2
H  .9 log2 .9  .1 log 2 .1
 .9( .152)  .1( 3.32)   .469
Interpretation of this index varies. If two species are
equally distributed, the measure of diversity is 1. If
there is little diversity, H is close to 0. In this case
H  0.5, so there is neither great nor little diversity.
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Slide 5.3-45