norm ebsary april 19, 2008 nsf msp spring 2008 pedagogy conference logs- powers, calculator,...

Norm Ebsary April 19, 2008

NSF MSP Spring 2008 Pedagogy Conference Logs- Powers, Calculator, GeoGebra, Slide Rule1

NSF MSP Spring 2008 Pedagogy Conference

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Logs- Powers, Calculator, GeoGebra,

Slide Rule

Norm Ebsary NSF MSP Spring 2008 Pedagogy ConferenceApril 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule2

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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.


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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.


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Introduction to Logs

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), andLog (1,000,000) = 6


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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

– is an irrational number like


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Logs of Small Numbers

0.0001 = 10-4, and Log(0.0001) = -4Numbers <1 have negative logarithms.

As the numbers get smaller and smaller, their logs approach negative infinity.

Logarithm is not defined for negative numbers.


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Numbers Not Exact Powers of 10

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


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Small Numbers Not Powers of 10

Log(0.001) = -3 and Log (0.0001) = - 4

What would be the logarithm of 0.0007?

– It should be between -3 and -4

In fact, Log (0.0007) = -3.155


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Calculator button marked LOG

N N Power of 10 Log (N)

1000 103 3.000

200 102.301 2.301

75 101.875 1.875

10 101 1.000

5 100.699 0.699


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Use Calculator for Table

N N Power of 10 Log (N)

1 100 0

.1 10-1 -1

.062 10-1.208 -1.208

.001 10-3 -3

.00004 10-4.398 -4.398


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Using GeoGebra with Logs

Log(1) = 0

Log(10) = 1


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Exponential to Log Forms

When y = bx

The log equivalent isLogby = x


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Graphing Logs in 3 easy steps

1. Invert log into Exponential Form

2. Inverse of Exponential form

3. Table convenient y values,

calculate x


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Graphing Logs Example

1. Invert log to Exponential

y = log2x y = 2x

2. Inverse in Exponential y = 2x x = 2y

3. Table convenient y values, calculate x

x y

1/4 -2

1/2 -1

1 0

2 1

4 2


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Slide Rule

http://www.ies.co.jp/math/java/misc/slide_rule/slide_rule.html


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Slide Rule Log Scales


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Example with 2x3 = 6


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Example with 6/3 = 2


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Example with 2x3 = 6


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Example with 6/3 = 2


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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?

The [H+] of the apple is 5.0 10– 4.

The [H+] of the banana is 6.3

10– 6.The apple has a higher concentration of hydrogen ions, so it is more acidic.

Apple

pH = –log[H+]

3.3 = –log[H+]

log[H+] = –3.3

[H+] = 10–3.3

5.0 10– 4

[H+] = 10–5.2

pH = –log[H+]

5.2 = –log[H+]

log[H+] = –5.2

Banana

6.3 10– 6

Log Example with Acid Levels


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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

L2 = 10 log

l1l0l2l0

Log Example with Sound (dB)


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L1 – L2 = 10 logl1l0

l2l0

– 10 log Find the decrease in loudness L1 – L2.

= 10 logl1l0

0.40l1l0

– 10 log Substitute l2 = 0.40l1.

= 10 logl1l0

– 10 log 0.40

•

l1l0

Product Property= 10 logl1l0

– 10 ( log 0.40 + log

)

l1l0

= 10 logl1l0

– 10 log 0.40 – 10 logl1l0

Distributive Property

= –10 log 0.40 Combine like terms.

4.0 Use a calculator, decrease in loudness of about 4 decibels.

Log Example with Sound (dB)


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The End

Questions?

norm ebsary april 19, 2008 nsf msp spring 2008 pedagogy conference logs- powers, calculator,...

Documents

slide rule slide

logs of small numbers

ln natural logs lnx

irrational number

relation number

given positive number

common logarithm

log e x