The Two-Place Logarithm Table: An Aid to Understanding and Use of Logarithms It is fairly disconcerting to learn that one of the greatest barriers to student understanding of chemical equilibrium calculations, such as are involved in pH and buffer calculations, is not of chemical origin but rather s. failure t o understand logarithms well enough to be able to use and mani~ulatethem with ease. Most of us relearn this dismal fact every school term and grumbie about the shortcomings of high school mathematics training. For thelast ten years I have undertaken to correct the situation in mv classes in a manner which mav he of interest to other teachers. Pricorresponds to the addition and subtraction of the related logarithm. 0.30). One could Hence the log of 4(= 2 X 2 or 2 9 is 0.60 (0.30 0.30); of 8( = 4 X 2 or 2') is 0.90 (0.60 continue in multiples of 2, i.e., 16, 32, 64, 128, 256, and 512, dividing each of these by 10or 100 (subtracting 1 or 2 from the log) to obtain values of the rest of the points. Thus,
+
1.28 2' Number
1
5 0
1.60 2" 10
2.00
+
2.56 28 100
2'
3.2 2& 10
4
2%
5.12 2' 100
6.4 2' 10
8
z5
10
It will be readily recognized that some of the numbers added toward the end of the series do not correspond exactly ) ~ worth worrying to the logarithm. The wont of these however, log 5.12 = 0.70, isinerror by only 0.01 ( ~ 2 %hardly about in a Zplace table! Even this can be readily corrected by using 0.70 = log 10 - log 2 = log 5; 0.40 = log 5/2, 0.10 = log 5/4. This adjusted table now reads Number
1
1.25
5 4
1.60
Z4 10
2.00 21
2.50
3.20
4.00
5 -
26 10
2'
2
5.00 10 2
6.40
26 10
8.00
10.0
2'
To handle numben thnr fnll between those whose logarithms nre tnbulnred, the s~udenris n s k d r u npproximnte this lo the nearrsr hundredth. It's x rare student whose guess is in error by mure than OU", even at first. "Egehalling" interpolations is far less mysterious to students than the formally correct technique. The advantages of using the development of this table as a student exercise and refresher are fairly substantial. First, the student easily and quicMy reviews the basic logarithmic principle (add logs when multiplying numbers, divide when subtracting). Second, without spending much time or effort the student soon has effectively memorized the Zplace table, making it possible for him to solve problems readily and reliahly (usually better than 5%). Inasmuch as the usual pH measurement is not really more reliable than 0.02 units, it follows that almost no pH calculation based on acid-base equilibria requires better than a two-place logarithm table. Indeed, this generally applies to all ionic equilibrium calculations. In my classes and exams, the Zplece log table has been used exclusively for a number of years not only with chemistry and chemical engineering majors, but with pharmacy, microbiology, and other health-related majors with good success and fine student acceptance. Also, there seems to be good grounds for believing that the use of the %dace log table in classroom work carries over into better ~erformancein the laboratory calculat~onswhere mastery of formal lGgarithm techniques are required
Volume 49, Number 5, May 1972
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325
