Transcript Ch 4. Indices and logsx

Essential Mathematics for Economics and Business, 4 th Edition
CHAPTER 4 : INDICES AND LOGS.
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© John Wiley and Sons 2013
© John Wiley and Sons 2013
Review indices
Relationship between indices and logs
Logs: The log of a number….
Rules for Logs
Some Worked Examples
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The rules for indices
Rule 1: To multiply numbers
with the same base, add the
indices.
a m  a n  a mn
Example by
Indices rule
Same example by
arithmetic
5 2  53  5 2  3
 55
(5.5.5)(5.5)  55
Rule 2: To divide numbers
with the same base, subtract
the index of the divisor.
m
a
mn

a
an
Rule 3: To raise an
exponential to a power,
multiply the indices.
(a )  a
m k
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mk
34
4 2

3
32
 32
3 3 3 3 3 3

3 3
1
 32
(2 3 ) 2  2 32
(2  2  2)(2  2  2)
2
6
= 26
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The relationship between indices and logs
number = basePower
N  bx
log base (number) = Power
log b N  x
A base is always base
logs are powers
Index form:
N  bx
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Log form:
log b N  x
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On the calculator….
Logs to base 10
log10N is written as logN
log1020 is written as log20 =1.3010
Logs to base e
logeN is written as lnN
loge20 is written as ln20 =2.9957
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The rules for logs
Rule 1: Add
log b ( M )  log b ( N )  log b ( MN )
Rule 2: Subtract.
M 
log b ( M )  log b ( N )  log b  
N
Rule 3: Log of an exponential
log b ( M ) z  z log b ( M )
Example
log10 (20)  log10 (4)  log10 (80)
1.3010  0.6021  1.9031
 20 
log e (20)  log e (4)  log e  
 4 
2.9957  1.3863  1.6094
log10 (8) 2  2 log10 (8)
1.8062  2(0.9031)
Rule 4: change of base rule.
log e (10) 
log new ( M )
log b ( M ) 
log new (b)
2.3026 
Note:log10(10)=1
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log10 (10)
log10 (e)
☺
☺
☺
☺
1
 2.3026
0.4343
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The relationship between indices and logs
To go from index to log
number = base Power
log base (number) = Power
A base is always base
To go from log to index reverse the process
log base (number) = Power
number = basePower
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© John Wiley and Sons 2013
The relationship between indices and logs
To go from index to log
number = base Power
log base (number) = Power
A base is always base
To go from log to index reverse the process
number = basePower
log base (number) = Power
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© John Wiley and Sons 2013
Solve equations using the rules for indices and logs
WE 4.17(a)
31  10
log(31)  x
Go from log
x
to index
1.4914  x
Alternatively
31  10 x
Take logs
log(31)  log(10 )
 x log(10)
1.4914  x
x
of each
side
Note:log10(10)=1
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Solve equations using the rules for indices and logs
WE 4.21(a)
20  3(1.08 x )
20
 1.08 x
3
log(20 / 3)  log(1.08 x )
log(20 / 3)  x log(1.08)
Divide
each side
by 3
Take logs
of each
side
log(20 / 3)
x
log(1.08)
0.8239
 x  24.65
0.0334
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Solve equations using the rules for indices and logs
WE 4.22(a)
log( x  2)  2.5
Go from log
to index
x  2  102.5
x  2  316.22777
x  316.22777  2
x  314.22777
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Solve equations using the rules for indices and logs
WE 4.22(b)
2 ln( x)  ln( x  1)  0
ln( x ) 2  ln( x  1)  0
 x2 
  0
ln
 x 1
Go from log to index
 x  0

  e  1
 x 1
2
x2  x 1
x2  x 1  0
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Solve the
quadratic
x  0.618
x  1.618
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