In the Classroom
Helping Students Make Sense of Logarithms and Logarithmic Relationships Ed DePierro and Fred Garafalo* School of Arts and Sciences, Massachusetts College of Pharmacy and Health Sciences, Boston, MA 02115; *
[email protected] Rick Toomey Department of Chemistry and Physics, Northwest Missouri State University, Maryville, MO 64468
Students often encounter difficulties when attempting to create or interpret mathematical representations of physical phenomena (1). Such problems are compounded when logarithms are involved, yet can remain hidden because of the ease with which logs are generated with an electronic calculator. This paper summarizes difficulties that students have consistently exhibited when using logarithms in classes taught by the authors, and offers approaches that instructors can draw on in order to actively address such difficulties. The latter have arisen through many student–instructor interactive sessions conducted during classes, help sessions, and laboratories (1). Sample problems based on these approaches are given in the online supplement. Difficulties Exhibited by Students Addressing students’ misconceptions or gaps in knowledge begins with identifying them. Our experience indicates that students commonly exhibit difficulties in the following areas.
1. Students are often unable to translate equations with the general form, logb N = L or antilogb L = N, into statements that do not contain the words, “log”, or “antilog”.
2. Often, students are unclear about the meaning of a base raised to a decimal exponent, possibly since there appears to be no simple way to determine the result other than by pressing a button on a calculator.
3. When translating expressions into logarithmic form, students may be able to invoke appropriate identities, yet often only by rote.
4. Students usually produce the incorrect number of digits when attempting to maintain significant figures as they determine logarithms.
5. Students do not realize that when a logarithm of a number carrying a unit is determined, the result carries no label.
6. Although students may be able to use equations in which one physical quantity is related to the logarithm of another, they are often unclear on the nature of such dependence.
7. If asked to represent data using a different base (e.g., 10 rather than e), some students attempt to change experimental data instead of using the appropriate conversion factor.
8. Use of the irrational number e (2.71828...) as the base for natural logarithms (ln), is a mystery to students. The relationship, x
± 1
1 d x ln x x
is fundamental in describing many situations in the physical sciences, however advanced students often have no clear picture of why the integral equals ln x. 1226
With first-year students, many of these difficulties arise from an inability to equate logarithms with exponents, as well as a lack of basic mathematics skills. For more advanced students, an inability to translate the symbolic representations of calculus into concrete operations is also a contributing factor. Helping Students Overcome Their Difficulties Experience suggests that both first-year and advanced students need opportunities to review basic mathematics skills, while learning to connect properties exhibited by a physical system with the corresponding mathematical representation. Examples in the online supplement elaborate on the approaches suggested in the following paragraphs. The numbers in the subsequent headings correspond to students’ difficulties listed above. Difficulty 1 Asking students to translate expressions into sentences that do not contain the words “logarithm” or “antilogarithm”, reinforces the idea that logarithms are exponents. For example, a student who can translate the expression log10 368 = ? as “To what exponent must 10 be raised to get the number 368?,” demonstrates clearer understanding than one who can only say “What is the base ten log of 368?” Similarly, when confronted with ln e‒kt = ?, a student who can only generate the answer, ‒kt, by symbol manipulation (moving ‒kt in front of ln e and then invoking the fact that ln e = 1) does not have as firm a foundation as one who understands that the question asks for the exponent to which e must be raised in order to produce e–kt. For expressions such as antilog10 ‒3 = ?, asking for the antilog of the exponent, ‒3, cues students to the role that the number ‒3 plays. It can also be noted that the answer, 10‒3, appears explicitly in the expression. The simplest case in which logarithms are used is in expressing quantities that vary over many orders of magnitude, such as H+ concentration in aqueous solutions.1 Going from [H+] to pH is usually less problematic for students than the reverse process. Having students write out the steps for the latter explicitly, until the process becomes familiar, is very helpful. For example, pH 3. 12 log10 H log10 H
H
3. 12 3. 12
10 3. 12 M 7. 6 t 10 4 M
Asking them to translate each equation into a sentence— and to rewrite the third step in a way that contains the word “antilog”—discourages rote symbol manipulation.
Journal of Chemical Education • Vol. 85 No. 9 September 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
In the Classroom Table 1. Data for a Hypothetical First-Order Process with a Half-Lifetime of Seven Minutes log1/2 ([A]/[A]0)
ln ([A]/[A]0)
log10 ([A]/[A]0)
0 = 7 × 0
t /min
1.0 = (1/2) 0
[A]/[A]0
0
‒0.693 × 0
‒0.301 × 0
7 = 7 × 1
0.5 = (1/2) 1
1
‒0.693 × 1
‒0.301 × 1
14 = 7 × 2
0.25 = (1/2) 2
2
‒0.693 × 2
‒0.301 × 2
21 = 7 × 3
0.125 = (1/2) 3
3
‒0.693 × 3
‒0.301 × 3
Difficulty 2 Students are usually familiar with square roots, so when the log10 3.162 is entered into a calculator, asking them to interpret the result, 0.5, is a means of introducing exponents that are decimals. A base-ten logarithm can be created for any number by summing various roots of that base (2). Such activities help students become more comfortable with noninteger exponents and reinforce important identities. For example, when the equation, (3.16)(1.78) = 5.62, is recast as (100.500)(100.250) = (100.750), comparison provides an opportunity for students to see that log10 5.62 should be 0.750, and reinforces the idea that exponent addition lies behind the phrase “log of a product is the sum of the logs” even when the exponents are not integers. Difficulty 3 Quantities expressed in scientific notation provide opportunities for students to reinforce their understanding of logarithmic identities and significant figure assignments. For example, students can be challenged to select from a group of similar distractors (without using a calculator) the expression 4 – log 7.6, as being equivalent to the pH of a 7.6 × 10‒4 M H+ solution. Such practice makes it more likely that students will understand why, for example, the equation A = A0e‒kt translates into ln [A] = ln [A]0 + ln e‒kt, when put into logarithmic form. Difficulty 4 When asked, for example, to give log10 957 to the correct number of significant figures, most students incorrectly report 2.98. When expressed as 9.57 × 102, however, it is easier for students to accept that the answer is 2.981, since the character “2” merely indicates that the number is at least 100 yet less than 1000. In general, the log can be reported to the same number of places beyond its decimal point as there are significant figures in the original number (3). For example, log10 (1.01 × 102) = 2.00432, although it is sufficient to report 2.004. The antilog of 2.004, when rounded, regenerates2 1.01 × 102. Difficulty 5 A formal discussion of the dimensionless nature of logarithms of labeled quantities has appeared in this Journal (4–6). Our experience suggests that students benefit from a less formal approach. Often, the logarithm of a ratio of quantities with the same label is taken (see difficulty 6, below), so in that case it is easy to argue that the result is dimensionless. In general, it can be pointed out that generating the logarithm of a number returns only part of that number, the exponent, and ignores the base. Therefore it is not surprising that the logarithm of a quantity (a number with a unit) ignores the unit. In “log10 35.6 M = 1.551” the value 1.551 is unitless. Only when the log is used with the base 10 is the concentration obtained: 101.551 M.
Difficulties 6 and 7 When one quantity is related to the logarithm of another quantity, data can be displayed in a way that helps clarify such a relationship. For example, when discussing first-order reaction kinetics, data displayed as in the first two columns of Table 1 can be used to emphasize the relationship between repeated time interval and exponent of the concentration ratio. Starting with [A]0 = 1 M simplifies the discussion. Column 3 can then be used to emphasize the point that exponents are logarithms, and that clock reading is directly proportional to the logarithm of the concentration ratio. The data for base 1/2 in column 3 can be recast as shown in column 4 for base e: e‒0.693(n) = (1/2)n, for n = 0, 1, 2, 3; or recast as shown in column 5 for base 10: 10‒0.301(n) = (1/2)n. Factors used for converting between bases e and 10 arise from ratios of the numbers in columns 4 and 5. Discussion centered on data in Table 1 reinforces the idea that raw experimental data do not change, but that such data can be processed in different ways. Setting ln ([A]/[A]0) = ‒kt then allows one to solve for the value of k, 0.099 min‒1, because base e is normally used with kinetics data. Further examples, using different time intervals and fractional concentration ratios other than multiples of ½, are needed before students appreciate the general nature of the arguments. The Table 1 display complements graphs of [A] versus t and ln [A] versus t that are usually presented. It also reinforces the idea that for first-order processes, the fraction of reactant lost in a given time interval is constant, not the rate of reaction. Students often mistake linearity in ln [A] versus t plots as indicating a constant reaction rate.3 For more advanced students, the connection between logarithms and rates can be reinforced through the ln [A] versus t plot. The starting point for a calculus-based treatment of firstorder processes involves rearranging the equation to give
d < A> k < A> dt
d < A> k d t
and then integrating both sides to yield
(1)
(2) ln < A> ln 0 k t Formal proof that equation 1 integrates to equation 2 is lengthy (7). However, when pointing out that ln [A]0 is the intercept, and ‒k is the slope, on the ln [A] versus t graph corresponding to eq 2, it can also be stressed that ‒k = dln [A]/dt, which rearranges to
d ln k d t
© Division of Chemical Education • www.JCE.DivCHED.org • Vol. 85 No. 9 September 2008 • Journal of Chemical Education
(3) 1227
In the Classroom
graph provides insight into the origin of this relationship. For a given value of x, a narrow rectangle with fixed width dx, centered at x, and height, 1/x, has an area of dln x. As x increases, and 1/x decreases, the decreasing rectangle area corresponds to a smaller error associated with ln x. Works by Maor (7), Pagel (8), and Salem, Testard, and Salem (9) provide further insights and references about logarithms; the latter represents collaboration between a chemist and a mathematician. Works by Fenn (10) and Bockhoff (11) provide interesting approaches to maximum work produced by an ideal gas. A recent article by the authors describes an approach used to introduce logarithmic dependence of electrochemical cell potentials (12).
1 x
0 e1
1
e
e2
x
Figure 1. Illustration of equal areas under the curve of y = 1/x. Each shaded segment has an area of 1 unit. Unit areas continue from e2 to e3, e3 to e4, etc., and e−1 to e−2, etc. Abscissa segments for values smaller than e−1 are not illustrated to maintain appropriate scale.
Comparing eq 1 to eq 3 is then a good way to informally reinforce the fact that d[A]/[A] = dln [A]. Difficulty 8 A graph of 1/x versus x, shown in Figure 1, is a convenient way to introduce the number e. This anticipates its use in more advanced discussions yet avoids lengthy formal proofs involving calculus (7) or reference to convergence of infinite series (7). Progressing along the x axis starting at x = 1 (in either direction), there is a relationship between the areas of segments under this curve and the number e. For any value of x, the area under the curve from 1 to x equals the base e logarithm of x. For example, going from 1 to 2, the area is 0.693 units, and e raised to the exponent 0.693 equals 2. Asking students to produce on a calculator the base e logarithm of the powers of any number, while the instructor refers to abscissa values and areas on the graph, serves to reinforce the relationship. (Students can be convinced that areas under the curve—e.g., from 1 to 2, 2 to 22, 22 to 23, etc.—are equal, by cutting out sections of a plot and weighing them.) These activities can help students see that the strange number e is related to something they usually have some knowledge of—quantities that are inversely proportional to one another. For more advanced students, such investigations can be used to reinforce the formal way this relationship is expressed through the symbolic language of calculus: x
1
± x dx 1
ln x
This equation and the Figure 1 graph have direct application in physical chemistry, for example, when discussing maximum work associated with the isothermal volume change of an ideal gas.4 The graph can also reinforce discussion of error propagation of logarithmic quantities. For a quantity x with uncertainty ∆x, the absolute error associated with ln x depends on the relative error in x: %x % ln x x When a vanishingly small uncertainty, dx, is considered, the 1228
Notes 1. Concentrations are usually used in introductory courses, rather than activities. 2. An activity in the supplement demonstrates that this approach works most of the time when numbers with positive or negative exponents are converted to base 10 or base e logarithms. Occasionally, small errors result when the original number is regenerated from the antilog. 3. When a plot of ln concentration vs t is nonlinear, tangent lines still represent fractional rates of change. 4. Returning to the kinetics example, the Figure 1 graph provides an unusual way to show that the same fraction of A is lost in a repeated time interval. If [A]/[A]0 is plotted on the abscissa, segments from 1 to ½, ½ to ¼, etc. all have areas = 0.693 = kt.
Acknowledgment The authors want to thank the reviewers for the many helpful suggestions they provided. Literature Cited 1. Cohen, J.; Kennedy-Justice, M.; Pai, S.; Torres, C.; Toomey, R.; DePierro, E.; Garafalo, F. J. Chem. Educ. 2000, 77, 1166–1173. 2. Feynman, R.; Leighton, R.; Sands, M. The Feynman Lectures on Physics; Addison Wesley: Reading, MA, 1965; Vol. 1, Ch. 22. 3. Jones, D. J. Chem. Educ. 1972, 49, 753–754. 4. Boggs, J. J. Chem. Educ. 1958, 35, 30–31. 5. Copley, G. J. Chem. Educ. 1958, 35, 366–367. 6. Mills, I. J. Chem. Educ. 1995, 72, 954–956. 7. Maor, E. e: The Story of a Number; Princeton University Press: Princeton, NJ, 1994. 8. Pagel, H. J. Chem. Educ. 1959, 36, 238–241. 9. Salem, L.; Testard, F.; Salem, C. The Most Beautiful Mathematical Formulas; J. Wiley and Sons, Inc.: New York, 1992. 10. Fenn, J. Engines, Energy, and Entropy; W. H. Freeman: San Francisco, CA, 1982; Ch. 4. 11. Bockhoff, F. J. Chem. Educ. 1962, 39, 340–342. 12. Toomey, R.; DePierro, E.; Garafalo, F. Chem. Educat. 2007, 12, 67–70.
Supporting JCE Online Material
http://www.jce.divched.org/Journal/Issues/2008/Sep/abs1226.html Abstract and keywords Full text (PDF) Links to cited JCE articles Supplement Problem sets (and answers) addressing the difficulties described A lesson on the maximum work done by the isothermal expansion of an ideal gas, also with questions and answers
Journal of Chemical Education • Vol. 85 No. 9 September 2008 • www.JCE.DivCHED.org • © Division of Chemical Education
