### 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.
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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
Ch
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.
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
3
), cubic feet (ft
3
), and so forth all having the same
dimension of cubic length [ L]
3
. 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
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:
g
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
A
2
=B þ B
2
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
1
2
gt
2
ð2Þ
where v = gt and s =1/2gt
2
, is numerically correct, this equation is
devoid of physical meaning (1), unless the dimension or unit for
68 Journal of Chemical Education
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Vol. 88 No. 1 January 2011
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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 -
E
a
RT

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
b
x if x ¼ b
y
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 -
E
a
RT
ð6Þ
But, in fact, it should have been written as (14)
ln
k
k
0
¼ ln
A
k
0
-
E
a
RT
ð7Þ
where k
0
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
f
0
/(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
E
a
¼ -R
dln k
dð1=TÞ
¼ -RT
2
dln k
dT
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
dT
¼
1
k
3
dk
dT
ð10Þ
where the dimensions of k are cancelled and which is equivalent
to dln(k/k
0
)/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
a
, log IC
50
, and So Forth. Mills has emphasized that the
correct definition of pH is (7, 16, 17):
pH ¼ - log
10
½H
þ
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
a
) or the
50% inhibitory concentrations and so forth. The argument of
the logarithm of the partition coefficient (for example, P
o/w
=
C
o
/C
w
) is already dimensionless because it is a ratio of concen-
trations.
r 2010 American Chemical Society and Division of Chemical Education, Inc.
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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
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
2
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
0
= yu, then eq 12 becomes
(4x yu)=(x yu)
2
, which simplifies to x =4/yu, a solution
that yields a perimeter equal to 16/yu but a surface equal to
16/y
2
u
2
.
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
2
ð13Þ
Solving for x:
x ¼ 4u
If we repeat the analysis with careful tracking of the dimensions,
we get
4x½L¼fx½Lg
2
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
0
½L
- 1
ð15Þ
where x
0
is a pure dimensionless number. If we now substitute x
0
in eq 12 instead of x we get
4x
0
¼ x
0
2
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
whenaunitisscaledareknownascomplete 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
2
2
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
dx
þ
Dx
2
2
3
d
2
f ðxÞ
dx
2
þ
Dx
3
6
3
d
3
f ðxÞ
dx
3
þ :::
¼ f ðxÞþ
X
¥
n ¼ 1
Dx
n
n!
3
d
n
f ðx Þ
dx
n
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
n
/1) (1/dx
n
) 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
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Vol. 88 No. 1 January 2011
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In the Classroom
in the arguments of these function. This distinction must be
clearly made to students of physical sciences early in their
Conclusions
Dimensional analysis is not generally given the prominence
it deserves in under graduate education. We have paid particular
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
considerably. The authors also thank Professors Katherine
V. Darvesh and Michael Bowen (both at Mount Saint Vincen t
University) and Mr. Hugo Bohorquez (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
Foundation for Innovation (CFI), and Mount Saint Vincent
University for financial support.
Literature Cited
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2. Menzel, D. H. Mathematical Physics; Dover Publications, Inc.:
New York, 1961.
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11. White, M. A. J. Chem. Educ. 1998, 75, 607609.
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p=1279091#post1279091 (accessed Oct 2010).
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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. 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). 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.