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Transcript
```Logarithms
Tutorial to explain the nature of
logarithms and their use in our
courses.
What is a Logarithm?
• The common or base-10 logarithm of a
number is the power to which 10 must be
raised to give the number.
• Since 100 = 102, the logarithm of 100 is
equal to 2. This is written as:
Log(100) = 2.
• 1,000,000 = 106 (one million), and
Log (1,000,000) = 6.
Logs of small numbers
• 0.0001 = 10-4, and Log(0.0001) = -4.
All numbers less than one have negative
logarithms.
• As the numbers get smaller and smaller,
their logs approach negative infinity.
• The logarithm is not defined for negative
numbers.
Numbers not exact powers of 10
• Logarithms are defined for all positive
numbers.
• Since Log (100) = 2 and Log (1000) = 3,
then it follows that the logarithm of 500
must be between 2 and 3.
• In fact, Log(500) = 2.699
Small Numbers not exact powers
of 10
• Log(0.001) = -3 and Log (0.0001) = - 4
• What would be the logarithm of 0.0007?
Since it is between the two numbers above,
its logarithm should be between -3 and -4.
• In fact, Log (0.0007) = -3.155
Why Logarithms?
• In scientific applications it is common to compare
numbers of greatly varying magnitude. Direct
comparison of these numbers can be difficult.
Comparison by order of magnitude using logs is
much more effective.
• Time scales can vary from fractions of a second to
billions of years.
• You might want to compare masses that vary from
the mass of an electron to that of a star.
• The following table presents an example:
Years before present (YBP)
Formation of Earth
4.6 x 109 YBP
Dinosaur extinction
6.5 x 107 YBP
First hominids
2 x 106 YBP
4
Last great ice age
1 x 10 YBP
First irrigation of crops
6 x 103 YBP
Declaration of Independence 2 x 102 YBP
Establishment of UWB
1 x 10 YBP
Data plotted with linear scale
5.E+09
4.E+09
3.E+09
2.E+09
All except
the first two
data points
are hidden
on the axis.
1.E+09
D
W
B
U
in
os
au
rs
H
om
in
id
s
Ic
e
Ag
e
Irr
ig
at
In
io
de
n
pe
nd
en
ce
0.E+00
Ea
rt h
Years before present
Events from Table I
Use Logs of Ages
• Because the data spans such a large range,
the display of it with a linear axis is useless.
It makes all events more recent than the
dinosaurs to appear the same!
• Instead, plot the logarithm of the tabular
data. Now the range to be plotted will be
much smaller, and the plot will distinguish
between the ages of the various events.
Log (YBP)
EVENT
YBP
Log(YBP)
Formation of Earth
4.6 x 10
9
9.663
Dinosaur extinction
6.5 x 107
7.813
First hominids
2 x 106
6.301
Last great ice age
1 x 104
4.000
First irrigation of crops
6 x 103
3.778
Declaration of Independence 2 x 102
2.301
Establishment of UWB
1.000
1 x 10
Plot using Logs
Events from Table I
Log(YBP)
10
8
All data are well
represented despite
their wide range.
6
4
2
0
th
r
Ea
D
e
rs
ds
g
i
u
A
a
in
s
e
o
Ic
om
in
H
n
ce
it o
n
e
a
d
g
i
n
e
Irr
ep
d
In
U
B
W
Your calculator should have a button marked
LOG. Make sure you can use it to generate
this table.
N
N as power of 10 Log (N)
1000
3
10
3.000
200
102.301
2.301
1.875
75
10
1.875
10
101
1.000
5
0.699
10
0.699
Also make sure you can use it to generate this
table.
N
1
N as power of 10 Log (N)
0
0
-1
-1
10
0.1
10
0.062
10-1.208
0.001
-3
10
0.00004 10-4.398
-1.208
-3
-4.398
Antilogs?
• The operation that is the logical reverse of
taking a logarithm is called taking the
antilogarithm of a number. The antilog of a
number is the result obtained when you
raise 10 to that number.
• The antilog of 2 is 100 because 102=100.
• The antilog of -4 is 0.0001 because 10-4 = 0.0001
Find the antilog function on your
calculator.
• To take antilogs, your calculator should
have one of the following:
• A button marked LOG-1
• A button marked 10x
• A button marked ALOG
• A two-button sequence such as INV
followed by LOG.
Make sure you can use your calculator to
generate this table.
N
As a power of 10 Antilog(N)
3
3
10
1000
1.5
101.5
31.62
1
1
10
10
0
100
1
-2
-3.4
10
-2
0.01
10
-3.4
0.0003981
Also make sure you can use it to generate this
table.
N
1
N as power of 10 Log (N)
0
0
-1
-1
10
0.1
10
0.062
10-1.208
0.001
-3
10
0.00004 10-4.398
-1.208
-3
-4.398
Natural Logarithms