Fermat's Library | Can One Take the Logarithm or the Sine of a Dimensioned Quantity or a Unit? Dimensional Analysis Involving Transcendental Functions annotated/explained version.

### TL;DR The correct way to convert the sentence "A is measured ...

It is not uncommon to find expressions such as "log temperature" in...

ahhhhh

In case of the exponential function, if we assume that the logarith...

Another way to look at the problem of taking the logarithm of a dim...

Terrence Tao has also a really good blogpost on the [mathematical f...

r 2010 American Chemical Society and Division of Chemical Education, Inc.

pubs.acs.org/jchemeduc

Vol. 88 No. 1 January 2011

Journal of Chemical Education 67

10.1021/ed1000476 Published on Web 10/13/2010

In the Classroom

Can One Take the Logarithm or the Sine of a

Dimensioned Quantity or a Unit? Dimensional

Analysis Involving Transce ndental Functions



erif F. Matta*

Department of Chemistry and Physics, Mount Saint Vincent University, Halifax, NS, Canada B3M 2J6

and Department of Chemistry, Dalhousie University, Halifax, NS, Canada B3H 4J3

*cherif.matta@msvu.ca

Lou Massa

Hunter College and the Graduate School, City University of New York (CUNY), New York,

New York 10021, United States

Anna V. Gubskaya

Department of Chemistry and Physics, Mount Saint Vincent University, Halifax, NS, Canada B3M 2J6

Eva Knoll

Faculty of Education, Mount Saint Vincent University, Halifax, NS, Canada B3M 2J6

Dimensional analysis (1) is central in the physical sciences

because it ensures both a deeper under standing of the physics and

allows a detection of dimensionally inconsistently derived equa-

tions, by requiring the two sides of an equality to carry the same

dimensions (a necessary but not sufficient condition for the

correctness of a derived equation). This analysis is not typically

taught in the systematic manner it deserves in undergraduate

courses nor does it feature prominently in standard physical

or analytical chemistry textbooks. Several authors, recognizing

the importance of dimensional analysis, devote a chapter or an

appendix to the topic, for example, the opening chapter of the book

Mathematical Physics by Donald Menzel entitled “Physical Dimen-

sions and Fundamental Units” (2).Thediscussion,however,usually

focuses on equations that seldom include transcendental functions

such as logarithms, exponentiation, trigonometric, and hyperbolic

functions.

In this article, the words dimensions and units are often used

interchangeably because their behavior in dimensional analysis

is identical, a unit being a dimension “dressed” with a given

magnitude, or to quote Hakala, a dimension is “a generalization

of a unit of measurement. That is, the dimensions of a physical

quantity are concerned with the nature and not the amount of the

quantity” (3). For example, volume may be expressed in cubic

meters (m

), cubic feet (ft

), and so forth all having the same

dimension of cubic length [ L]

. Different units are rela ted by a

constant dimensionless number . So while there is a conceptual

distinction between a unit and a dimension, their interchange is

tantamount to changing the symbol of a given physical quantity,

since the dimensionless number that converts one unit to another

does not appear in dimensional analysis.

There has been a vigorous debate in the literature for over six

decades about the fate of dimensions (and units) when quantities

with dimensions are the arguments of logarithms (or exponentiation)

(4-10, see also ref 11). The same issue has also been a subject

of discussion and disagreement recently on the Internet (12).

Some of these views are correct, some are wrong, and some

are correct for the wrong reasons. The goals of this article are

to (i) summarize the issue, (ii) expose some common traps and

misconceptions, and (iii) generalize the discussion to include

transcendental functions other than logarithms and exponentiation,

which were not discussed before (for example, trigonometric and

hyperbolic functions).

Bridgman's Principle of “Dimensional Homogeneity”

The rule of dimensional homoge neity needs to be intro-

duced early in the education of students of the physical sciences:

quantities A and B on both sides of the addition and subtraction

operators as well as equality and inequality must have the same

dimensions, that is, in each one of the following expressions A

and B are dimensionally homogeneous:

A þ B

A - B

A ¼ B

A > B

A < B

etc:

ð1Þ

Rule 1 is a generalization of the property of the operator

expression dim (A ( B)=dimA = dim B showing that dimensional

operator algebra differs, for example, from the algebra of numbers, as

discussed in this Journal (13).AlthoughitisacceptabletowriteA þ

=B þ B

for dimensionless (pure) numbers, this expression is

meaningless if A and B representphysicalquantitiesthatare

essentially the product of a pure number and a physical dimension

expressed in a system of units (1, 14). Thus, although

v þ s ¼ gt þ

ð2Þ

where v = gt and s =1/2gt

, is numerically correct, this equation is

devoid of physical meaning (1), unless the dimension or unit for

68 Journal of Chemical Education

Vol. 88 No. 1 January 2011

pubs.acs.org/jchemeduc

r 2010 American Chemical Society and Division of Chemical Education, Inc.

In the Classroom

each term, including the coefficients, is defined and, further, that these

are compatible under the applied operations according to rule 1.

This is a statement of the familiar expression that one cannot add,

subtract, or compare apples and oranges. Dimensional analysis is the

method used to ensure dimensional homogeneity by making explicit

the dimensional validity of an equation by ascribing, when necessary,

the correct dimensionality to constants of proportionality (2).The

application of these rules can be useful in spotti ng errors in a derived

equation. Conversely, their satisfaction, although necessary, is not

sufficient to guarantee the correctness of an equation.

Apparent Problems in Dimensional Analysis Involving

Transcendental Functions

By definition, transcendental functions such as logarithm

(to any base), exponentiation, trigonometric functions, and

hyperbolic functions act upon and deliver dimensionless numer-

ical values. We now explore some common examples of mis-

conceptions involving these functions and the resolution of such

misconceptions.

Logarithms

Arrhenius Equation. If the argument of the function is already

dimensionless as, for example, the argument of the exponential

function in the Arrhenius equation

k ¼ A exp -



ð3Þ

then, according to rule 1 both k and the pre-exponential factor A

must necessarily have the same dimensions (of reciprocal time)

because the exponential factor is dimensionless. A difficulty arises

when one has to insert a quantity with dimension (expressed in a

given system of units) in the argument of a transcendental function.

The logarithm is defined as the reciprocal operation of exponentia-

tion (12),thatis

y ¼ log

x if x ¼ b

ð4Þ

where b, x,andy are real numbers, b being the base of the logarithm.

This definiti on precludes the association of any physical dimension

to any of the three variables, b, x,andy, so as not to violate rule 1.

Consider the following example discussed by Molyneux (8)

logð10 gramsÞ¼logð10  gramÞ

¼ logð10Þþlogðgram Þ

¼ 1 þ logðgramÞ

ð5Þ

where the log is to the base 10. Molyneux's suggestion that the

last equality defines the meaning of log 10 gram is mistaken for

the following reason. If we follow this suggestion, then if we

apply definition in eq 4 one may ask: what is the exponent y

(a number) to which one should raise the base b, that will yield

gram(s)? Because there are no reasonable answers to this ques-

tion, the quantity (log gram) is meaningless as is eq 5 as a whole.

The notion that, in logarithmic relationships, dimensions are

carried additively (15) is a misconception: dimensions are not

carried at all in a logarithmic function.

To linearize the Arrhenius equation (eq 3), it is often modified

by taking the logarithm of both sides:

ln k ¼ ln A -

ð6Þ

But, in fact, it should have been written as (14)

¼ ln

ð7Þ

where k

is the unit rate constant in the chosen system of units.

In this manner, the dependence of the logarithm on the unit

used is irrelevant because the argument of the logarithm in eq 7 is

a ratio of two variables having the same unit, that is, a pure

number. The method of expressing the arguments of the

logarithms in the Arrhenius equation shown in eq 7 is recom-

mended when tabulating or plotting data as it avoids potential

ambiguities. In this manner, the numerical value of a (dimensioned)

physical quantity is disengaged from its dimension, such that (11)

physical quantity=unit ¼ numerical value ð8Þ

as, for example, in ΔH

/(kJ mol

-1

)=-285.9. This is known as

quantity calculus, the importance of which has been underscored

some time ago in this Journal and has been adopted as the Journa l 's

conventional style (11).

The differential form of the Arrhenius equation is

¼ -R

dln k

dð1=TÞ

¼ -RT

dln k

ð9Þ

which is another deceptive form that may give the appearance of

taking the logarithm of a dimensioned quantity, that is, the rate

constant k. Formulas of this form occur in abundance in physical

sciences, and the resolution of this apparent problem is in the

identity (1)

dln k

ð10Þ

where the dimensions of k are cancelled and which is equivalent

to dln(k/k

)/dT, that is, taking the logarithm of a dimensionless

number.

In addition to logarithms , it is equally meaningless to include

dimensioned quantities as the arguments of trigonometric or

hyperbolic functions because these are defined as ratios (the sine

of an angle is the ratio of the length of the opposite side to the

length of the hypotenuse, the cosi ne is the ratio of the length

of the adjacent side to the length of the hypotenuse, etc.) The

hyperbolic functions, themselves defined in terms of either expo-

nential or trigonometric functions, cannot operate on quantities to

which physical dimensions are attached either.

pH, pK

, log IC

, and So Forth. Mills has emphasized that the

correct definition of pH is (7, 16, 17):

pH ¼ - log

½H



mol dm

- 3

ð11Þ

Thus, before taking the logarithm, one has to divide the concen-

tration by the unit of concentration used to obtain a dimension-

less or unitless number. A similar division by appropriate units is

necessary before applying the negative of the logarithm to

quantities such as the acid dissociation constant (K

) or the

50% inhibitory concentrations and so forth. The argument of

the logarithm of the partition coefficient (for example, P

o/w

) is already dimensionless because it is a ratio of concen-

trations.

r 2010 American Chemical Society and Division of Chemical Education, Inc.

pubs.acs.org/jchemeduc

Vol. 88 No. 1 January 2011

Journal of Chemical Education 69

In the Classroom

Relative versus Absolute Units as Arguments of Transcen-

dental Functions

Using a calculator, what happens to the dimensions or units

when one inserts a dimensioned quantity as an argument of a

transcendental function? To develop an answer with the student

asking such a question, we may ask them the following question:

What is the length of the side of a square whose perimeter is

equal in magnitude to its area? An incorrect approach is to

write:

4x ¼ x

ð12Þ

where x is the length of the side, because if we solve for x then any

square of side length = 4 would satisfy eq 12, but 4 of what units?

In other words, any square could be a solution to this problem,

provided the unit is defined as one-fourth of its side length. The

mistake lies of course in ignoring the dimensions or units.

Instead, if the original unit of length u is changed to one that

is y times the original unit, that is, u

= yu, then eq 12 becomes

(4x  yu)=(x  yu)

, which simplifies to x =4/yu, a solution

that yields a perimeter equal to 16/yu but a surface equal to

16/y

Applying rule 1 to the above problem helps reaching the

correct answer. To balance the units or dimensions according to

rule 1, the equation needs to be rewritten as

4u  x ¼ x

ð13Þ

Solving for x:

x ¼ 4u

If we repeat the analysis with careful tracking of the dimensions,

we get

4x½L¼fx½Lg

4 ¼ x½Lð14Þ

where the quantity in square bracket [L] refers to the dimension

of length (or any of its surrogate units). Rule 1 demands that both

sides of eq 14 have the same dimensions, and because the left-

hand side is a pure number, so must be the quantity on the right-

hand side (but it is not). Therefore

x ¼ x

½L

- 1

ð15Þ

where x

is a pure dimensionless number. If we now substitute x

in eq 12 instead of x we get

¼ x

ð16Þ

an equation that appears identical to eq 12 but with a profound

difference: eq 16 re-expresses eq 12 in relative dimensionless form

and will always be true. Equations similar to 16 that are unchanged

whenaunitisscaledareknownas“complete equations” (1).

Atomic units that are often used in molecular and atomic

physics and quantum chemistry are relative (dimensionless) units

because they provide the numerical coefficients of the rati o of the

quantity of interest to a standard reference value (18). Other

dimensionless units include radian, percent, mole, bit, and so

forth.

In optics, to describe a sinusoidal disturbance as a function

of distance from a chosen or igin, one must first multiply x[L] by a

“propagation number”, k[L]

-1

, before inserting in the argument

of the sine function (19). Thus,

ψðx, tÞ¼A sin kðx - vtÞð17Þ

where ψ is the disturbance in specified dimensions and units

(e.g., electric or magnetic field strength), A is the maximum

disturbance or the amplitude in the same units as the distur-

bance, and sin k(x - vt) is dimensionless and provides the

temporal and spatial periodicity of the disturbance. In the words

of Hecht (19):

It's necessary to introduce the constant k simply because we cannot

take the sine of a quantity that has physical units. The sine is the ratio

of two lengths and is therefore unitless. Accordingly, kx is properly in

radians, which is not a real physical unit [since it is a ratio].

Thus, whenever a quantity is to be inserted into the argument of

a transcendental function it is either dimensionless from the start

or it has to be made dimensionless. In other words, dimensions

must be left at the doorstep of transcendental functions.

The Fallacy of an Argument Based on Taylor's Expansion

Wikipedia is a site that is gaining wide popularity among

students and faculty, and it and other such online publications

can no longer be ignored. Although generally very useful, some-

times these nonpeer-reviewed online publications can perpetuate

wrong (but seemingly plausible) views and analyses. An example

of a flawed argument that nevertheless reaches the correct conclu-

sion is Wikipedia 's entry, as of September 2010, and which

dismisses the insertion of a dimensioned quantity into the

argument of a transcendental function on the basis of the Taylor

expansion of these functions (20). The Wikipedia article argues

that the Taylor expansion of transcendental functions leads to

terms of differ ent dimensions that cannot be added or subtracted

(rule 1), and hence, the argument can not be dimensioned. The

example quoted is (20) :

lnð3kgÞ¼3kg-

9kg

þ ::: ð18Þ

The Taylor expansion written in eq 18 is incorrect and deceptive.

If we write the formal Taylor expansion,

f ðx þ DxÞ¼ f ðx ÞþD x

df ðxÞ

f ðxÞ

þ :::

¼ f ðxÞþ

n ¼ 1

f ðx Þ

ð19Þ

eq 19 shows that, should f (x) be dimensioned, then every term in

the expansion has the same dimensions as f(x), because the

dimensions of (∂x

/1)  (1/dx

) cancel as discussed in ref 14.

Therefore, the addition (or subtraction) of the terms in a Taylor

expansion is numerically and dimensionally permissible and the

equation satisfies dimensional homogeneity (rule 1). The same

argument carries over to all transcenden tal and algebraic func-

tions that can be expanded as Taylor series. The reason for the

necessity of including only dimensionless real numbers in the

arguments of transcendental function is not due to the dimen-

sional nonhomogeneity of the Taylor expansion, but rather to

the lack of physical meaning of including dimensions and units

70 Journal of Chemical Education

Vol. 88 No. 1 January 2011

pubs.acs.org/jchemeduc

r 2010 American Chemical Society and Division of Chemical Education, Inc.

In the Classroom

in the arguments of these function. This distinction must be

clearly made to students of physical sciences early in their

undergraduate education.

Conclusions

Dimensional analysis is not generally given the prominence

it deserves in under graduate education. We have paid particular

attention in this article to dimensional analysis when the

equations involve transcendental functions because this is per-

haps one of the least frequently discussed topics in dimensional

analysis. When a physical quantity is to be inserted into the

argument of a transcendental function, care must be taken to

convert them first into dimensionless ratios.

We have reviewed common misconceptions and errors

associated with dimensional analysis involving these functions,

and in particular an example of a deceptive argument, related

to Taylor expansion, that leads to correct conclusions but for

the wrong reasons. The fallacy of the latter argument promul-

gated online underscores the importance that care be exerted

when using nonpeer-reviewed online publications in under-

graduate education. The importance of a thorough under-

standing of dimensional analysis for future scientists, who will

derive as well as apply formulas and functions, cannot be over-

emphasized.

Acknowledgment

The authors are grateful to the excellent suggestions made

by three anonymous reviewers that helped to improve this article

considerably. The authors also thank Professors Katherine

V. Darvesh and Michael Bowen (both at Mount Saint Vincen t

University) and Mr. Hugo Bohorquez (Ph.D. Candidate,

Dalhousie University) for useful discussions and suggestions.

L.M. thanks the City University of New York CUNY-PSC fund

for grant support. C.F.M. acknowledges the Natural Sciences and

Engineering Research Council of Canada (NSERC), Canada

Foundation for Innovation (CFI), and Mount Saint Vincent

University for financial support.

Literature Cited

1. Bridgman, P. W. Dimensional Analysis; Yale University Press:

New Haven, 1931.

2. Menzel, D. H. Mathematical Physics; Dover Publications, Inc.:

New York, 1961.

3. Hakala, R. W. J. Chem. Educ. 1964, 41, 380–384.

4. DePierro, E.; Garafalo, F.; Toomey, R. J. Chem. Educ. 2008, 85,

1226–1228.

5. Freeman, R. D. J. Chem. Educ. 1997, 74, 899.

6. Molyneux, P. J. Chem. Educ. 1995, 72, 955–956.

7. Mills, I. M. J. Chem. Educ. 1995, 72, 954–955.

8. Molyneux, P. J. Chem. Educ. 1991, 68, 467–469.

9. Boggs, J. E. J. Chem. Educ. 1958, 35,30–31.

10. Copley, G. N. J. Chem. Educ. 1958, 35, 366–367.

11. White, M. A. J. Chem. Educ. 1998, 75, 607–609.

12. Physics Forum, What is the Unit after you take the Natural

Logarithm? http://www.physicsforums.com/showthread.php?

p=1279091#post1279091 (accessed Oct 2010).

13. Musulin, B. J. Chem. Educ. 1964, 41, 622.

14. Berberan-Santos, M. N.; Pogliani, L. J. Math. Chem. 1999, 26,

255–261.

15. Denker, J. S. Dimensionless Units, http://www.av8n.com/physics/

dimensionless-units.htm. post1279091 (accessed Oct 2010).

16. Mills, I.; Cvitas, T.; Homann, K.; Kallay, N.; Kuchitsu, K. Quan-

tities, Units and Symbols in Physical Chemistry, the IUPAC Green

Book, 2nd ed.; Blackwell Scientific: London, 1988.

17. Cohen, E. R.; Cvitas, T.; Frey, J. G.; Holmstrom, B.; Kuchitsu, K.;

Marquardt, R.; Mills, I.; Pavese, F.; Quack, M.; Stohner, J.; Strauss,

H. L.; Takami, M.; Thor, A. J. Quantities, Units and Symbols in

Physical Chemistry, the IUPAC Green Book, 3rd ed.; RSC Publish-

ing, Cambridge: London, 2007.

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to Advanced Electronic Structure Theory; Dover Publications, Inc.:

New York, 1989.

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ahhhhh ### TL;DR The correct way to convert the sentence "A is measured in metres" into mathematics is to write the dimensionless quantity "A/(1 m)" In this paper, the authors pay particular attention to dimensional analysis when the equations involve transcendental functions like the logarithm, exponential or sine by exposing some common misconceptions. Notice that you took ln(2 km) to be the area under 1/x from x=1 km to x=2 km. Then if you convert to m, your limits of integration are not x=1 m to x=2000 m. We were originally starting from 1 km = 1000 m, so instead, our new limits are from x=1000 m to x=2000 m. The integral from 1000 to 2000 of 1/x is indeed still 0.69=ln(2). Terrence Tao has also a really good blogpost on the [mathematical formalisation of dimensional analysis](https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/). It is not uncommon to find expressions such as "log temperature" in many physics textbooks. In case of the exponential function, if we assume that the logarithm is dimensionless then $$ \ln (e^{2 km}) \text{ is dimensionless} $$ Now, if we exponentiate both sides we get that $e^{2 km}$ is a dimensionless number, meaning the km unit was lost in this process. We thus conclude that the exponential function is "blind" to the unit you put there, be it a km or a m, which again makes no sense. Another way to look at the problem of taking the logarithm of a dimensioned quantity is by analyzing how the logarithm is defined. The natural logarithm ln X is defined as the area under the curve y =1/x from x=1 to X. $$ \ln X = \int_{x=1}^X \frac{1}{x} dx $$ This integral is the sum of an infinite number of dimensionless slices of area "dx/x". With this in mind, the ln (2 km) is the area under the curve from x=1 km to x = 2 km - ln(2) = 0.69. If we apply the same reasoning and calculate the ln (2000 m) we end up with ln(2000) = 7.6 which doesn't make sense as we were expecting that ln (2 km) = ln(2000 m).

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