TANTON S TAKE ON LOGARITHMS CURRICULUM TIDBITS FOR THE MATHEMATICS CLASSROOM APRIL 01 Try walking into an algera II class and writing on the oard, perhaps even in silence, the following: = Your turn! ( 8) 5( 5) = 1 7 = = = 0.01 0 7 0 = 16 6 1 7 = 7 = 9 = ( million) = = 1 = 00 = 0.1 1 6 = _ 6 5 = 1 I et your students, perhaps after some initial mumling amongst themselves, will start to correctly fill in the lanks. Great! You have just taught logarithms! Congratulate your students for their cleverness, and then say: Unfortunately, for unusual historical reasons, we don t use the word. We use the strange word logarithm, which we write as log for short. Now go though the list on the oard and cross out each word and replace it with log. Taking the time to do this in a showy way rings home the point that logarithms are just s. Whenever we see the word log we are to think. log is just of that gives the answer., the Now finish the lesson y recounting the historical story detailed net!
THE HISTORICAL MOTIVATION for LOGARITHMS During the Renaissance in Europe the study of science really took off. Scholars were doing eperiments and collecting data left, right and center. With the invention of the telescope (Galileo first called the device a perspicillum) astronomers recorded the positions of heavenly ojects and tracked their motions with higher and higher levels of precision. Geographers were using trigonometry and making measurements of land features on the Earth. But advances in all this glorious work were severely held ack y one very annoying issue: all calculations had to e done y hand (no calculators in the 1500s) and paper-and-pencil arithmetic is slow and tedious! Adding data values (say, to calculate the average of ten eperimental results) is not too onerous. (Could you compute.1+.097+.87 fairly easily y hand?) But multiplying multi-decimal numers, as one might need to do for scientific formulas, is mighty laorious! (Care to work out.1.097.87 without a calculator?) It seemed totally asurd to have significant advances in science held ack y the simple fact that performing multiplication y hand is so slow and laorious (and very much open to making errors!). complicated way to the numer M, and the second oject a velocity which varied in a way related to the numer N and to the distance the first car still needed to travel along the line. He found that in his strange method computing the ratio of the velocities of the two ojects had, in essence, converted the multiplication prolem M N into a calculation of addition. (Napier also changed the scale of this prolem so that the numer 7 =, 000, 000 would play a prominent role. He felt this would assist geographers and astronomers who were often dealing with large numers.) Napier ased the name for his technique on the Greek words logos for ratio and arithmos for numer to get logarithm. Napier s methods were tremendously successful and highly praised, despite eing so complicated and hard to understand in the theory. So that scholars would not have to worry aout conceptual details Napier, with the help of Henry Briggs (1561-160), pulished tales of logarithm values. This made the computation of M N etraordinarily straightforward: 1. Look up the values log M and log N in the tale.. Add the two numers.. Look ack at the tale and see which entry has log M + log N as its value. This entry is the product M N. So in the late 1500s a Scottish mathematician y the name of John Napier (1550 1617) took it upon himself to help his eleaguered scientific colleagues. He set out to devise a method that would turn multiplication prolems into addition prolems. He succeeded in this task, ut his approach was highly creative to say the least! To find a product M N Napier envisioned two ojects moving along a section of a straight line. He gave the first oject a velocity related in some It wasn t until almost a century later that scholars realized that Napier s logarithms are actually very simple and already familiar: they are nothing more than eponents computed ackwards! But, of course, y then the word logarithm was firmly in place and the name stays with us to this day. (With asolutely no disrespect to Napier, life would e so much easier for students today if we let go of the work logarithm and used the word.) Comment: Henry Briggs suggested to Napier he ase his logarithms on the numer. This, after all, is the numer on which
our arithmetic system. Today ase-ten logarithms are sometimes called Briggsian Logarithms. They are also called common logarithms and the suscript ten is usually omitted in their use. (Thus, for instance, log 7 is understood to mean log 7.) By the way In today s notation, Napier s logarithms would e written: 7 N log 1 1 7. You can see now why it 7 was so hard to recognize Napier s method as simple eponentiation in reverse!) LOGARITHMS REALLY DO DO THE TRICK! Let s estalish first some asic properties of logarithms. Understanding their validity is simply a matter of pure thought! Recall: ( ) ( ) log =! PROPERTY 1: log ( ) = 1 Reason: The of that gives the answer is oviously one! PROPERTY : log (1) = 0 Reason: The of that gives the answer 1 is 0. PROPERTY : log ( ) = Reason: The of that gives the answer is oviously! PROPERTY : log ( ) (This one throws people!) = Reason: Recall that log is the of that gives the answer. So if we use it as a of, we must otain the answer! operations (as mathematicians in the 1700s finally noticed!) PROPERTY 5: NAPIER S DREAM log ( N M ) = log N+ log M. Logarithms convert multiplication prolems into addition prolems. EXAMPLE: Consider log ( 8 ). This, y definition, is the of that gives the answer 8. Now, 8 is a product of three twos ( log (8) = ) and is the product of five twos ( log () = 5 ). Thus 8 is a product of + 5 twos. We have: log 8 = + 5= log (8) + log () This eample offers a hint that logarithms do indeed accomplish what Napier set out to do. Although this eample is helpful, we still need a formal proof of this property that works for all numers, not just on convenient whole numer eponents. Formal Reason: We want to prove that log M N log M + log N. ( ) is That is, we want to show that log M + log N is the right of to use to get the answer M Let s see if it is. N. log M+ log N log M log N M N = =. (Here we used the eponent rule a+ a =, and then we used prop..) YES indeed! log ( M) log ( N) + is the of that gives the answer M N! COMMENT: Properties and show that eponentiation and taking logarithms undo each other. They are inverse
EXERCISE: I tell you that, for some numer : log = 0.69 log = 1.098 log 5= 1.609 Without a calculator, find log, log 6 log 8, log 9, log and log 600. Estimate log 7 and log 70. Also estimate log( 1 5 6 7 8 9 )., EXERCISE: Estalish the rule N log = log N log M M Comment: As division is the reverse process of addition, it is not surprising that sutraction appears here. Although property 5 was the reason for the invention of logarithms in the year 1600, this need is no longer relevant for today. We multiply numers with calculators! However, there is a sith property of significant importance to us still. PROPERTY 6: log ( M ) log ( M) =. EXAMPLE: log ( M ) = log( M M M M) = log( M) + log( M) + log( M) + log( M) = log ( M) Of course, this approach relies on the convenience of whole numer eponents. Formal Reason: Property 6 claims that the of we need to get the answer M is log ( M ). Let s check! (We ll again use eponent rules and property.) log M log M = ( ) = M. M is indeed ( M ) YES! log EXERCISE: Which numers are suitale ases for logarithms? M ever make sense? Does log1 Does log0 log ( ) =. Does log ( M) M ever make sense? always make sense? log M always make sense? Does Does 0.000000 log ( M ) always make sense? Does log 1 19 M always make sense? EXAMPLE: Solve = 7 5 + Answer: Property numer 6 shows how to ring eponents down from an epression. Let s apply a logarithm to oth side of the equation. We can use any ase of our choice. Let s use ase- logarithms (as they appear on our calculators!). We have: + ( ) = ( ) + + = + log log 7 5 log log log 7 log 5 log + ( + )log = log 7 + ( )log 5 log + log log5= log 7 log 5 log yielding log 7 log 5 log = log + log log5 log 7 log 5 log16 = log + log log15 log 7 / 00 = log 1 /15 EXAMPLE: How do I work out log 5 (7) on my calculator? Answer: Let s play with it. If = log5 7, then is the of 5 that gives the answer 7 : 5 = 7.
5 We have now an epression with no mention of logarithms. Let s apply a logarithm of our choice, the common logarithm, to each side of the equation. This gives: log 5= log 7 log 7 =. log 5 I can put this into my calculator! Comment: Please don t memorise a change of ase formula. Like all things in math, JUST DO IT! INTERNET RESEARCH: Find the details of Napier s ratio of velocities. VIDEOS: Here are some short videos recapping the history and some of the mathematics of logarithms. Enjoy! /?p=55 www.jamstanton.com/?p=556 01 James Tanton tanton.math@gmail.com