Norm Ebsary April 19, 2008 NSF MSP Spring 2008 Pedagogy Conference Logs- Powers, Calculator, GeoGebra, Slide Rule 1 NSF MSP Spring 2008 Pedagogy Conference Podcasting Logs Logs- Powers, Calculator, GeoGebra, Slide Rule

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 2 John Napier logarithm (lŏg' ə rĭth ə m) [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.

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 3 Why use Logarithms? Scientific applications common to compare numbers greatly varying sizes. Time scales can vary from a nano-second (10 -9 ) to billions (10 9 ) of years. You could compare masses of an electron to that of a star.

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 4 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 = 10 2, the logarithm of 100 is equal to 2. Written as: Log(100) = 2 1,000,000 = 10 6 (one million), and Log (1,000,000) = 6

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 5 Introduction to Logs So a common logarithm is log 10 ( x) = log(x) There are also natural logarithms – which are referred to as ln Natural logs ln(x) = log e (x) Remember e = – is an irrational number like 

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 6 Logs of Small Numbers = 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.

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 7 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

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 8 Small Numbers Not Powers of 10 Log(0.001) = -3 and Log (0.0001) = - 4 What would be the logarithm of ? – It should be between -3 and -4 In fact, Log (0.0007) =

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 9 Calculator button marked LOG

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 10 Use Calculator for Table

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 11 Using GeoGebra with Logs Log(1) = 0 Log(10) = 1

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 12 Exponential to Log Forms When y = b x The log equivalent is Log b y = x

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 13 Graphing Logs in 3 easy steps 1. Invert log into Exponential Form 2. Inverse of Exponential form 3. Table convenient y values, calculate x

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 14 Graphing Logs Example 1.Invert log to Exponential y = log 2 x  y = 2 x 2.Inverse in Exponential y = 2 x  x = 2 y 3.Table convenient y values, calculate x xy 1/4-2 1/

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 15 Slide Rule

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 16 Slide Rule Log Scales

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 17 Example with 2x3 = 6

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 18 Example with 6/3 = 2

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 19 Example with 2x3 = 6

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 20 Example with 6/3 = 2

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 21 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 –  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

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 22 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 l 1 = present intensity. Let l 2 = reduced intensity. Let L 1 = present loudness. Let L 2 = reduced loudness. Write: l 2 = 0.04 l 1 L 1 = 10 log L 2 = 10 log l1l0l1l0 l2l0l2l0 Log Example with Sound (dB)

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 23 L 1 – L 2 = 10 log l1l0l1l0 l2l0l2l0 – 10 log Find the decrease in loudness L 1 – L 2. = 10 log l1l0l1l0 0.40l 1 l 0 – 10 log Substitute l 2 = 0.40l 1. = 10 log l1l0l1l0 – 10 log 0.40 l1l0l1l0 Product Property = 10 log l1l0l1l0 – 10 ( log log ) l1l0l1l0 = 10 log l1l0l1l0 – 10 log 0.40 – 10 log l1l0l1l0 Distributive Property = –10 log 0.40Combine like terms. 4.0 Use a calculator, decrease in loudness of about 4 decibels. Log Example with Sound (dB)

Podcasting Logs Norm Ebsary NSF MSP Spring 2008 Pedagogy Conference April 19, 2008 Logs- Powers, Calculator, GeoGebra, Slide Rule 24 The End Questions?