Logarithms and Exponentials

A familiarity with logarithms and exponentials is quite important to mastery of many scientific disciplines at the college level, especially chemistry. An exponential function is one whose argument is a power of some number called the base. For example, when we use scientific notation, we use a base 10 exponential function. When we read the exponential part of a number written in scientific notation in terms of "power of ten" or "order(s) of magnitude", we are in fact computing the result of the logarithmic function, which is the inverse of the exponential function. (Note that an exponential is often called an "antilogarithm".) Thus, we write the base 10 exponential of 2 as 10² = 100. The inverse function, a base 10 logarithm in this instance, is written as log(100) = log(10²) = 2.

An important property of any logarithmic function is that

log ab  =  log a  +  log b

We can illustrate this with simple base 10 log and exponential functions. As we have seen, log(100) = 2 means that 10² = 100. Similarly, log(1000) = 3 means that 10³ = 1000. What is log (100 x 1000)? Well, 100 x 1000 = 100,000 or 10⁵. Apparently, log (100,000) = 5. But 5 = 2 + 3 = log (10²) + log(10³) = log (100 x 1000). Although we have considered an artificially simple example, it is important to remember that the above property is true for any numbers a, b and any base logarithmic function. Other similar properties, that follow from the above (and the definition of 1/x = x^–1)

log (1/a)  =  log a^–1 =  – log a     (a > 0)

log aⁿ  =  n log a     (a>0, n any number)

log (a/b)  =  log ab^–1  =  log a – log b     (a > 0, b > 0)

log a^mbⁿ  =  m log a  +  n log b         (a > 0, b > 0, m, n any numbers)

Bear in mind such properties any time you find yourself using log functions in problem-solving.

Common (Base 10) and natural logarithms

To be sure, when speaking of logarithms and exponentials, we ought to be careful to specify what base we are using. In other words, we specify the value of the constant A in the defining relationship

y  =  log_Ax   <=>   A ^y =  x

which can be translated into words as "y is equal to the base A logarithm of x if and only if x is equal to A raised to the y power". We attempt in this way to accommodate the general nature of logarithms and exponentials as a class of functions. The practical truth of the matter is that for the most part we only use base 10 logarithms and natural logarithms. In general chemistry, we are first introduced to base 10 logarithms, which are also called common logarithms. Let us explicitly define base 10 and natural logarithms (and their cognate exponentials or antilogs) and specify conventional notations for them as follows:

Common (Base 10) logarithms

y  =  log₁₀x   <=>   10 ^y =  x

We will adopt the convention that "log x", with the base unspecified, means common log, i.e. base 10.

Natural logarithms

y  =  log_ex   <=>   e ^y =  x

The adjective "natural" adheres to a base e logarithm because of how it arises naturally in mathematics. Consider how the number p arises naturally in plane geometry as the ratio of the circumference to the diameter of a circle. Likewise, e arises naturally - for example, in calculus where e is a unique number such that d/dx(e ^y) = e ^y. We will adopt a specific convention for the natural logarithm function, namely that "ln x" means natural log, i.e. base e = 2.71828.... .

Graphs: ln x, (for further development: log x, e^x, e^–x, )

The graph of the natural log (ln) function is shown below. The natural logarithm is usually written as ln, and we will try to stick to this convention, although a mathematician is apt to interpret "log" to mean natural log. There are several things to note about the function that the graph illustrates.

If the value y of the function is to be 1, then x must be equal to the base value,

e = 2.71828....

Like p, e is a transcendental number, both a real and an irrational number, a non-repeating decimal. ln x is not defined unless x > 0. For 0 < x < 1, ln x is negative, with ln x approaching negative infinity as x approaches zero.

When x = 1, it must be that y = 0. This would be true for a logarithmic function of any base, since the zeroth power of any number is defined as 1.

Finally, note how slowly y increases as x increases for x > 1. This shows that the ln function in particular (and any log function in general) compresses a large range of input (x) values into a narrow range of output (y) values. This is a very useful property for representation of data or quantities that vary over a huge range.

Logarithms and exponentials of other bases and interconversions

As noted above, the general definition for a logarithmic function of base A is

y  =  log_Ax   <=>   A ^y =  x

The next most encountered base is 2, which is closely related to binary numbers. In chemistry, we rarely if ever use logarithms other than common (base 10) or natural (base e) logarithms. We can readily develop a formula for interconversion of logarithms of any two bases, but the most common operation we carry out is to convert natural logarithms into base 10 logarithm which we can accomplish with the formula below:

ln x  =  (ln 10)(log x)  =  2.303 log x

ln 10  =  2.3025851

Here is a proof that the conversion factor for natural to base 10 log is ln 10. Let us label the factor we seek as f. Thus f log x = ln x for all x > 0. In particular, if x = 10

f log 10  =  ln 10  or   f = ln 10, since log 10 = 1.

More generally, we can prove that the conversion factor for base A log to base B is log_AB. Labeling the factor we seek as f, we can write f log_Ax = log_Bx for all x > 0. In particular, if x = B,

f log_BB  =  log_AB  or   f  =  log_AB, since log_BB  = 1.

Logarithmic functions in chemistry: pH, pK, pDiddy

The pH is a chemical measurement of acidity. The pH is defined as a logarithmic function of the hydronium ion concentration (abbreviated here as H⁺),

pH  =  – log [H⁺].

Note the minus sign in the definition creates an inverse relationship between [H⁺] and pH. Strongly acidic conditions very high [H⁺] means pH is very low. But in practice, [H⁺] varies over many orders of magnitude so the logarithmic scaling of pH makes it a useful quantity to define. Another example of such a quantity is K, the equilibrium constant for a reaction. We will want to be able to use pK values, where

pK  =  – log K

to quantify the equilibrium composition of physical and chemical systems. In acid-base chemistry, the quantity pK_a, defined as

pK_a  =  – log K_a  =  – log {[H⁺][A^– ] / [HA]}.

is used as a measure of the strength of acids on the same scale as the pH scale.

In the range of hydronium ion concentrations we will typically encounter, the pH ranges between 0 and 14. As we may well know already, a pH of 7 - that is, [H⁺] = 1.0 x 10^–7 M - is considered neutral, and pH less than 7 is considered acidic, while pH > 7 is basic. As a measure of basicity, and a counterpart to pH, we introduce pOH, defined as

pOH  =  – log [ OH ^– ].

A simple relationship exists between pH and pOH in aqueous solutions of acidic and basic compounds

pH  +  pOH  =  14

and its basis in the definition and value of the ionization constant K_w.

log, exp, and scientific units

You may have noticed that in the definition of pH above, it was mentioned that it is defined in terms of the logarithm of the hydronium ion concentration, but no further account of the actual units and what happens to them when we take the logarithm. Our unit of concentration is moles per liter (mol L^–1), which we also call molarity (M). In taking the logarithm, the units are dropped by dividing the concentration by a standard concentration of exactly one molar. In this way, any argument to a log or exp function can be treated as a relative, unitless number. Good thing, too, since it is hard to see how to make sense of something like "log(meters)".

(For further development) Treatment of significant figures in logs and exponentials

Some exercises

1. Calculate the value of (a) K_a if pK_a = 4.326 ; (b) pH for [H⁺] = 5.278 x 10^–6

2. Show that

pK₂  –  pK₁  =  DpK  =  p(K₂/ K₁)

Further Exercises :

3. True or false: pDiddy  =  – log (Sean Combs)

4. Compute e^–DG°/RT for DG° = 241.6 kJ mol^–1, T = 298.15 K, and R = 8.3145 J mol^–1 K^–1.

(Answer  e^{+ 92.26} = 1.2 x 10⁴⁰)

Further special topics

Entropy is a logarithmic function of probability

There is an intrinsic connection between the logarithmic function and entropy (ref 1). On the molecular level, entropy arises in a probabilistic or statistical treatment of the behavior of an astronomical collection of molecular size particles. The equilibrium state of an isolated system is the most probable state. "The increase in S as an isolated system proceeds toward equilibrium is directly related to the system's going from a state of low probability to one of high probability" (See ref. 1, p.96). Consider a system consisting of two subsystems, 1 and 2, isolated from one another (in other words, not interacting). Using the facts that entropy is both a state function and an extensive variable, we compute the total entropy of the system (both subsystems) as S = S₁ + S₂. Now if S is some function of the probability of the state f(p), and subsystems S₁ and S₂ are independent, then the probabilities of their states are independent, and the rules of probability say the probability of the state [1 + 2] of the system would be the product of the probabilities for state [1] of subsystem 1 and state [2] of subsystem 2. Therefore,

f(p[1+2])  =  f(p[1]p[2])  =  g(p[1]) + h(p[2])

Where f, g, and h are the functions that define the entropies of the system 1+2, subsystem 1, and subsystem 2, respectively. We might expect f, g, and h to be closely related functions if they are to all to correspond to entropy and satisfy the above criterion. We may have even guessed that log functions could fit the bill, recalling that log xy = log x + log y. In fact, it can be shown that f, g, and h must all be of the form

S  =  f(p)  =  k ln p + a

where k and a are both constants. One can go on to show that although the constant a can take on different values for the entropy function defined for different systems, k must be the same for all such entropy functions, and to estimate its magnitude (see p.97, Ref.1). The constant k is an important, fundamental constant of nature called the Boltzmann constant:

k  =  1.380669 x 10^–23 J K^–1.

References

1. Ira N. Levine Physical Chemistry, 5th edition, (McGraw-Hill, 2002)
2. Atkins P, de Paula J. Physical Chemistry for the Life Sciences (Oxford, 2006)