Transcript Properties of Logarithms Lesson 5.5 Basic Properties of Logarithms   Note box on page 408 of text Most used properties log(a  b)  log a 

Properties of
Logarithms
Lesson 5.5
Basic Properties of
Logarithms


Note box on page 408 of text
Most used properties
log(a  b)  log a  log b
a
log    log a  log b
b
log a n  n  log a
Using the Log Function for Solutions
100   0.886   25
t

Consider solving

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Previously used algebraic techniques
(add to, multiply both sides) not helpful
Consider taking the
log of both sides and
using properties
of logarithms


log 100   0.886   log 25
t
log100  log 0.886t  log 25
t  log 0.886  log 25  log100
log 25  log100
t
log 0.886
Try It Out

Consider solution of
1.7(2.1) 3x = 2(4.5)x
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Steps
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Take log of both sides
Change exponents inside log to coefficients outside
Isolate instances of the variable
Solve for variable
Natural Logarithms
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We have used base of 10 for logs
Another commonly used base for logs is e



)
e has other interesting properties
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e is an irrational number (as is
Later to be discovered in calculus
Use ln button on your calculator
Properties of the Natural
Logarithm
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Recall that y = ln x  x = ey
Note that
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ln 1 = 0 and ln e = 1
ln (ex) = x (for all x)
e ln x = x
(for x > 0)
As with other
based logarithms
ln( a  b)  ln a  ln b
a
l ln    ln a  ln b
b
n
ln a  n  ln a
Use Properties for Solving
Exponential Equations
1.04  3
t

Given

t
Take log of
log 1.04  log 3
both sides
Use exponent
t

log(1.04)

log3
property
Solve for what
log1.04
t
was the exponent



Note this is not the same as
log 1.04 – log 3

log 3
Misconceptions
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log (a+b) NOT the same as
log (a-b) NOT the same as
log (a * b) NOT same as
log (a/b) NOT same as
log (1/a) NOT same as
log a + log b
log a – log b
(log a)(log b)
(log a)/(log b)
1/(log a)
Usefulness of Logarithms
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Logarithms useful in measuring quantities which
vary widely
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Acidity (pH) of a solution
Sound (decibels)
Earthquakes (Richter scale)
Chemical Acidity
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pH defined as
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pH = -log[H+]
where [H+] is hydrogen ion concentration
measured in moles per liter
If seawater is [H+]= 1.1*10-8
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then
–log(1.1*10-8) = 7.96
Chemical Acidity
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What would be the hydrogen ion concentration
of vinegar with pH = 3?
Logarithms and Orders of
Magnitude
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Consider increase of CDs on campus since
1990
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Suppose there were 1000 on campus in 1990
Now there are 100,000 on campus
The log of the ratio is the change in the order of
magnitude
 100, 000 
log 
2
 1000 
Decibels
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Suppose I0 is the softest sound the human
ear can hear
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measured in watts/cm2
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And I is the watts/cm2 of a given sound
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Then the decibels of the sound is
 I 
10  log  
 I0 
The log of the
ratio
Logarithms and Orders of
Magnitude

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We use the log function because it “counts” the
number of powers of 10
This is necessary because of the vast range of
sound intensity that the human ear can hear
Decibels
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If a sound doubles, how many units does its
decibel rating increase?
Change of Base Formula
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We have used base 10 and base e
What about base of another number
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log 2 17 = ?
Use formula
Use base 10 or base e which
calculator can do for you
logb x
log a x 
logb a

Think how to create a function to do this on your
calculator
Assignment
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Lesson 5.5
Page 444
Exercises 1 – 85 EOO
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Assign change of base spreadsheet
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Due in 1 week.