Download Technology for Introductory Statistics

NSF MSP Spring 2008
Pedagogy Conference
Podcasting Logs
Logs- Powers, Calculator,
GeoGebra, Slide Rule
1
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
John Napier 1550 - 1617
logarithm (lŏg'ərĭthəm)
[Gr.,=relation number],
number associated with a
positive number, being the
power to which a third
number, called the base, must
be raised in order to obtain
the given positive number.
2
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Why use Logarithms?
Scientific applications common to
compare numbers greatly varying sizes.
 Time scales can vary from a nano-second
(10-9) to billions (109) of years.
 You could compare masses of an electron
to that of a star.

3
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Introduction to Logs



4
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. Written as: Log(100) = 2
1,000,000 = 106 (one million), and
Log
(1,000,000)
=
6
Norm Ebsary
NSF MSP Spring 2008 Pedagogy Conference
April 19, 2008
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Introduction to Logs
So a common logarithm is log10( x) =
log(x)
 There are also natural logarithms

–
which are referred to as ln
Natural logs ln(x) = loge(x)
 Remember e = 2.718281828

–
5
is an irrational number like 
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Logs of Small Numbers
0.0001 = 10-4, and Log(0.0001) = -4
Numbers <1 have negative logarithms.
 As the numbers get smaller and smaller,
their logs approach negative infinity.
 Logarithm is not defined for negative
numbers.

6
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Numbers Not Exact Powers of 10



7
Logarithms are for positive numbers only.
Since Log (100) = 2 and Log (1000) = 3,
then it follows that the logarithm of 500
must be between 2 and 3
The Log(500) = 2.699
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Small Numbers Not Powers of 10

Log(0.001) = -3 and Log (0.0001) = - 4

What would be the logarithm of 0.0007?
–

8
It should be between -3 and -4
In fact, Log (0.0007) = -3.155
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Calculator button marked LOG
N
10
3
3.000
200
10
2.301
2.301
75
101.875
1.875
10
101
1.000
1000
5
9
N Power of 10 Log (N)
Norm Ebsary
April 19, 2008
10
0.699
0.699
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Use Calculator for Table
N
1
0
N Power of 10 Log (N)
0
1
10
.1
10-1
-1.208
.062
10
.001
10-3
.00004 10
Norm Ebsary
April 19, 2008
-4.398
0
-1
-1.208
-3
-4.398
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Using GeoGebra with Logs
Log(1) = 0
Log(10) = 1
1
1
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Exponential to Log Forms
When y = bx
The log equivalent is
Logby = x
1
2
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Graphing Logs in 3 easy steps
1
3
1.
Invert log into Exponential Form
2.
Inverse of Exponential form
3.
Table convenient y values, calculate x
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Graphing Logs Example
1. Invert log to Exponential
x
y = log2x  y = 2
2. Inverse in Exponential
x
y
y=2  x=2
3. Table convenient y values,
calculate x
1
4
Norm Ebsary
April 19, 2008
x
1/4
1/2
1
2
4
y
-2
-1
0
1
2
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Slide Rule
1
5
http://www.ies.co.jp/math/java/misc/slide_rule/slide_rule.html
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Slide Rule Log Scales
1
6
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Example with 2x3 = 6
1
7
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Example with 6/3 = 2
1
8
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Example with 2x3 = 6
1
9
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Example with 6/3 = 2
2
0
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Log Example with Acid Levels
The pH of an apple is about 3.3 and that of a banana is about 5.2. Recall that the
pH of a substance equals –log[H+], where [H+] is the concentration of hydrogen
ions in each fruit. Which is more acidic?
Apple
Banana
pH = –log[H+]
3.3 = –log[H+]
log[H+] = –3.3
[H+] = 10–3.3
5.0

10– 4
The [H+] of the apple is 5.0  10– 4.
2
1
pH = –log[H+]
5.2 = –log[H+]
log[H+] = –5.2
[H+] = 10–5.2
6.3

10– 6
The [H+] of the banana is 6.3  10– 6.
The apple has a higher concentration of hydrogen ions, so it is more acidic.
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Log Example with Sound (dB)
Manufacturers of a vacuum cleaner want to reduce its sound intensity to 40%
of the original intensity. By how many decibels would the loudness be reduced?
Relate: The reduced intensity is 40% of the present intensity.
Define: Let l1 = present intensity. Let l2 = reduced intensity.
Let L1 = present loudness. Let L2 = reduced loudness.
Write: l2 = 0.04 l1
L1 = 10 log l1
l0
2
2
L2 = 10 log l2
Norm Ebsary
April 19, 2008
l0
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
Log Example with Sound (dB)
L1 – L2 = 10 log l1 – 10 log l2
l0
Find the decrease in loudness L1 – L2.
l0
= 10 log l1 – 10 log 0.40l1
l0
Substitute l2 = 0.40l1.
l0
= 10 log l1 – 10 log 0.40 • l1
l0
= 10 log l1 – 10
l0
l0
( log 0.40 + log
l1
l0
= 10 log l1 – 10 log 0.40 – 10 log l1
l0
l0
= –10 log 0.40
2
3
4.0
Norm Ebsary
April 19, 2008
)
Product Property
Distributive Property
Combine like terms.
Use a calculator, decrease in loudness of about 4 decibels.
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule
Podcasting Logs
The End
2
4
Questions?
Norm Ebsary
April 19, 2008
NSF MSP Spring 2008 Pedagogy Conference
Logs- Powers, Calculator, GeoGebra, Slide Rule