Note for Logarithm - L by Placement Factory

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Explaining Logarithms Common Logarithms of Numbers N 0 1 A Progression of Ideas Illuminating an Important Mathematical Concept log ( x / y) = log x – log y log ( x * y) = log x + log y 4 y=x 3 2 x y=b b>1 –3 –2 1 –1 logb b x = x y = logb x b>1 1 2 3 3 4 –1 –2 by = x is equivalent to y = logb x blogb x = x log b m = m log b By Dan Umbarger logp x = logq x logq p

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Dedication This text is dedicated to every high school mathematics teacher whose high standards and sense of professional ethics have resulted in personal attacks upon their character and/or professional integrity. Find comfort in the exchange between Richard Rich and Sir Thomas More in the play A Man For All Seasons by Robert Bolt. Rich: “And if I was (a good teacher) , who would know it?” More: “You, your pupils, your friends, God. Not a bad public, that …” In Appreciation I would like to acknowledge grateful appreciation to Mr. (Dr.?) Greg VanMullem, who authored the awesome freeware graphing package at mathgv.com that allowed me to communicate my ideas through many graphical images. A picture is truly worth 1,000 words. Also a big “Thank you” to Dr. Art Miller of Mount Allison University of N.B. Canada for explaining the “non-integer factoring technique” used by Henry Briggs to approximate common logarithms to any desired place of accuracy. I always wondered about how he did that! Four colleagues, Deborah Dillon, Hae Sun Lee, and Fred Hurst, and Tom Hall all graciously consulted with me on key points that I was unsure of. “Thank you” Paul A. Zoch, author of Doomed to Fail, for finally helping me to understand the parallel universe that we public high school teachers are forced to work in. “Thank you” Shelley Cates of thetruthnetwork.com for helping me access the www. And the biggest “Thank you” goes to John Morris of Editide (info@editide.us) for helping me to clean up my manuscript and change all my 200 dpi figures to 600 dpi. All errors, however, are my own. Copyright © 2006 by Dan Umbarger (Dec 2006) Revised, June 2010 Single copies for individuals may be freely downloaded, saved, and printed for non-profit educational purposes only. Donations welcome!!! Suggested donation $6 students ages 1-18, $12 adults 19 and above. See mathlogarithms.com. Single and multiple bound copies may be purchased from the author at mathlogarithms.com or Dan Umbarger 7860 La Cosa Dr. Dallas, TX 75248-4438

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Explaining Logarithms A Progression of Ideas Illuminating an Important Mathematical Concept By Dan Umbarger Brown Books Publishing Group Dallas, TX., 2006 John Napier, Canon of Logarithms, 1614 “Seeing there is nothing that is so troublesome to mathematical practice, nor doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances….Cast away from the work itself even the very numbers themselves that are to be multiplied, divided, and resolved into roots, and putteth other numbers in their place which perform much as they can do, only by addition and subtraction, division by two or division by three.” As quoted in “When Slide Rules Ruled” by Cliff Stoll, Scientific American Magazine, May 2006, pgs. 81

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Table of Contents Foreword............................................................................................................. ii Note to Teachers ................................................................................................. v Chapter 1: Logarithms Used to Calculate Products ............................................ 1 Chapter 2: The Inverse Log Rules ...................................................................... 9 Chapter 3: Logarithms Used to Calculate Quotients ........................................ 20 Chapter 4: Solving for an Exponent—The General Case ................................. 25 Chapter 5: Change of Base, e, the Natural Logarithm ...................................... 29 Chapter 6: “When will we ever use this stuff?” ............................................... 37 Chapter 7: More about e and the Natural Logarithm ........................................ 56 Chapter 8: More Log Rules .............................................................................. 66 Chapter 9: Asymptotes, Curve Sketching, Domains & Ranges ....................... 69 Chapter 10 … Practice, Practice, Practice ........................................................ 76 Appendix A: How Did Briggs Construct His Table of Common Logs? .......... 85 Appendix B: Cardano’s Formula—Solving the Generalized Cubic Equation . 93 Appendix C: Semilog Paper ............................................................................. 94 Appendix D: Logarithms of Values Less than One .......................................... 95 Appendix 2.71818: Euler’s Equation, An Introduction……………………...96 Appendix F: Exponents, Powers, Logarithms … What’s the difference?. . . 99 Answers: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 i