Download Lesson 2.5

Logarithmic Functions
Lesson 2.5
How to Graph These Numbers?
Distance from the Sun
Consider the
vast range
of the numbers
Object
Distance (million km)
Mercury
58
Venus
108
Earth
149
Mars
228
Jupiter
778
Saturn
1426
Uranus
2869
Neptune
4495
Pluto
5900
Proxima Centauri
4.1E+07
Andromeda Galaxy
2.4E+13
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How to Graph These Numbers?
Distance (million km)
3E+13
2.5E+1
3
2E+13
1.5E+1
3
1E+13
5E+12
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What's wrong
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How to Graph These Numbers?
What's wrong
with this
picture?
Distance (million km)
6000
5000
4000
3000
2000
1000
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We need a way to
set a scale that fits all
the data
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How to Graph These Numbers?
The solution:
Set the scale
to be the
exponent
of the distance
Distance (million km)
1E+14
1E+12
1E+10
1E+08
1E+06
10000
100
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This is called a
logarithmic scale
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A New Function
Consider the exponential function
y = 10x
Based on that function, declare a new
function
x = log10y
You should be able to see that these are
inverse functions
x
In general a  b  log b a  x
The log of a number
is an exponent
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The Log Function
Try These
log39 = ?
3 9
so log 3 9  2
2
log232 = ?
2  32
so log 2 32  5
5
log 0.01 = ?
Note: if no base
specified,
102  default
0.01 is
base of 10
so log 0.01  2
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Properties of Logarithms
Note box on page 105 of text
Most used properties
log( a  b)  log a  log b
a
log    log a  log b
b
n
log a  n  log a
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Change of Base Theorem
To find the log of a number for a base
other than 10 or e …
• Use
logb x
log a x 
logb a
Note new spreadsheet
assignment on
Blackboard
Where b can be any base
• Typically 10 or e
• Available on calculator
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Change of Base Theorem
Create a function for your calculator
• Define
function
• Try it
• Verify
10
Solving Log Equations
Use definition of logarithm
• Rewrite log equation as an exponential
equation
5
log 4 x 
2
Result
5
2
4 x
x = 32
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Solving Exponential Equations
Use property of logarithms
log a  n  log a
or
n
ln e  n
n
.01x
Consider 5e
9
• Isolate exponential expression
• Take ln of both sides
• Solve for x
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Doubling Time
What if inflation is at the 5% rate …
How long until prices double?
2  P  P  1.05 
t
Strategy
• Divide through by P
• Take log of both sides
• Bring t out as coefficient
• Solve for t
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Assignment
Lesson 2.5
Page 121
Exercises 1 – 79 EOO
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