Linear and Logarithmic Scales
Spring 2006 – Week 6
PHY 131
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Linear and Logarithmic Scales
moving one increment up
adds the same increment
moving
one
increment up
along the
scale
adds the same increment
moving
one
increment up
along the
scale
adds the same increment
movingthe
one
increment up
along
scale
adds the same increment
moving one increment up
along the scale
adds the same increment
along the scale
Spring 2006 – Week 6
PHY 131
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1
Linear and Logarithmic Scales
moving one increment
up multiplies by 10
moving one increment
up multiplies by 10
moving one increment
up multiplies by 10
moving one increment
up multiplies by 10
moving one increment
up multiplies by 10
Spring 2006 – Week 6
PHY 131
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Definition of Powers
ab is the number that results when you multiply
the number a with itself b times.
Examples:
23 = 2 x 2 x 2 = 8
55 = 5 x 5 x 5 x 5 x 5 = 3125
107 = 10 x 10 …. x 10 = 10,000,000
7 “10s” here
Spring 2006 – Week 6
PHY 131
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2
Rules for Powers
1
an
a = a × a × a... × a
a− n =
am × an = am + n
am
m−n
=
a
an
n
(a m )n = a m × n
Spring 2006 – Week 6
PHY 131
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Definition of Inverse Powers
(Roots)
a1/b is the number that, when multiplied
b times with itself, results in a.
Square
root of 2
Examples:
21/2
21/2 x 21/2 = 21/2+1/2 = 21 = 2
121/5
121/5 x 121/5…x 121/5 = 121 = 12
5 “121/5” here
Fifth root
of 12
Spring 2006 – Week 6
PHY 131
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3
Scientific Notation
Used to express very large and very small numbers
as a number between 0 and 1 multiplied by a power
of 10.
0.5315 x 107 = 5,315,000
(move decimal point +7 places to the right)
0.3715 x 10-4 = 0.00003715
(move decimal point –4 places to the left)
Spring 2006 – Week 6
PHY 131
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Fractional Powers
If 103 = 10 x 10 x 10, what does 103.4 mean?
Write exponent as ratio 3.4 =
Then 10
3.4
= 10
17
5
1
17 5
34 17
=
10 5
= (10 ) = (10 × 10 × ... × 10)
1
5
So 10 3.4 = fifth root of 100,000,000,000,000,000=2512
Spring 2006 – Week 6
PHY 131
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4
Fractional Powers
If 103 = 10 x 10 x 10, what does 103.4 mean?
Or we could break it up and write the
4
2
exponent as sum 3.4 = 3 +
= 3+
10
5
Then 10
3+
2
5
2
1
= 10 3 × 10 5 = 10 3 × (10 2 ) 5
So 10 3.4 is 1000 × the fifth root of 100 =1000 × 2.512 = 2512
Spring 2006 – Week 6
PHY 131
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Why do we need Logarithms?
100=1
10x = 2
101=10
What power of ten do we need to get 2?
To find x we need to define….
Spring 2006 – Week 6
PHY 131
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5
Definition of Logarithm
logbx is the number to which power b must
be raised to result in x.
b is called the base
Common bases are
10
e = 2.71828…
Spring 2006 – Week 6
(decade log or lg)
(natural log or ln)
PHY 131
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Examples
Note: We will always use the base 10 and will omit the base.
So instead writing log10x we will just write log 10.
Log 1000 = 3
Log 1 = 0
Log 0.01 = -2
Spring 2006 – Week 6
because 103 = 1000
because 100 = 1
because 10-2 = 0.01
PHY 131
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6
Rules for Logarithms
Spring 2006 – Week 6
PHY 131
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Rule 1
Log of a product = sum of logs
log(xy) = log x + log y
log(32) = log(8 × 4) = log 8 + log 4
Spring 2006 – Week 6
PHY 131
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7
Rule 2
Log of a ratio = difference of logs
x
log( ) = log x − log y
y
log(0.5) = log(1 / 2) = log1 − log 2
Spring 2006 – Week 6
PHY 131
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Rule 3
Log of an exponent = factor x log
log(x N ) = N log x
log(10 6 ) = 6 log(10)
Spring 2006 – Week 6
PHY 131
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8
Logs w/o a Calculator
Any number can be written in scientific notation as:
# x power of ten
Powers of ten are easy:
Number
between 0…1
log(10N) = N log10 = N
Thus to find the log( # x 10N) = log # + N, we just have to
know the logarithms of 1, 2, 3…..8, 9.
Spring 2006 – Week 6
PHY 131
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3 out of 9 is enough!
But when we use the logarithm rules, we get
away with memorizing the logs of only three
numbers:
log1 = 0
log 4 = 2 log 2 = 0.602
log 7 = 0.845
log 2 = 0.301
log 5 = 1 − log 2 = 0.699
log 8 = 3log 2 = 0.903
log 3 = 0.477
log 6 = log 2 + log 3 = 0.778
log 9 = 2 log 3 = 0.954
Spring 2006 – Week 6
PHY 131
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9
Some Examples
log(32) = log(8 × 4) = log 8 + log 4 = 0.903 + 0.602 = 1.505
log(50) = log(5 × 10) = log 5 + log10 = 0.699 + 1 = 1.699
log(0.5) = log(1 / 2) = log1 − log 2 = 0 − 0.301 = −0.301
log(0.08) = log(8 × 0.01) = log 8 + log 0.01 = 0.903 − 2 = −1.097
log(10 6 ) = 6 log(10) = 6
log(0.08) = log(8 × 10 −2 ) = log 8 + log10 −2 = log 8 − 2 log10 = −1.097
Spring 2006 – Week 6
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10