Logarithms and Exponentials

Exponentials and logarithms appear in many formulas in chemistry. We have already encountered them in the definitions of prefixes in Table 1.2, which are essentially a shorthand to avoid large powers of ten (we can write 17 ps instead of 1. 7 x 10-11 s). In addition to powers of 10, we frequently use powers of e = 2.7183. .. and occasionally use powers of 2. The number e (base of natural logarithms) arises naturally in calculus, for reasons we will discuss briefly later (calculus classes explain it in great detail). Powers of e occur so often that a common notation is to write exp(x) instead of e.  

---

The range of values of x for which each of the series is convergent is stated at the right of the series. 2.2.4.1 Exponential and Logarithmic Series... 

Remember that exponentials and logarithms have no units so that the units of this exponential must cancel.)... 

In this section, we take an approach that is characteristic of conventional perturbation theories, which involves an expansion of a desired quantity in a series with respect to a small parameter. To see how this works, we start with (2.8). The problem of expressing ln(exp (—tX)) as a power series is well known in probability theory and statistics. Here, we will not provide the detailed derivation of this expression, which relies on the expansions of the exponential and logarithmic functions in Taylor series. Instead, the reader is referred to the seminal paper of Zwanzig [3], or one of many books on probability theory - see for instance [7], The basic idea of the derivation consists of inserting... 

The exponential and logarithm functions have clear convex and concave characters, and do not do well in a nearly linear function. The last three functions provide minor corrections on the linear function, and do quite well. It should be pointed out that the first four functions have two arbitrary parameters each, the quadratic has three and the cubic has four. We expect that when the number of parameters increases, we can correlate ever more complicated data. For the cubic equation with three parameters and only six data points, there remain only three degrees of freedom. It is often said that Give me three parameters, and I can fit an elephant, so that it is a greater achievement to fit complicated data with as few parameters as possible, in the spirit of what is called the Occam s Razor. ... 

Applications of Exponentials and Logarithms Nuclear Disintegrations and Reaction Kinetics... 

The definition of many elementary functions can be extended to complex variables. Pol5momial, exponential, and logarithmic functions are discussed here. 

It is assumed that the exponential and logarithmic functions are thoroughly familiar to the reader as also are their relatives the circular and hyperbolic functions. However, the Bessel functions are introduced to the student much later and have less claim to familiarity. The exponential integral is also a function which occurs in several places and is worthy of some explanation. This appendix is therefore intended to provide a little background on the applications of these functions. 

The computation of a cyclone fractional or grade efficiency depends on cyclone parameters and flow characteristics of particle-laden gases. The procedure involves a series of equations containing exponential and logarithmic functions. Koch and Licht [ 12] described a cyclone using seven geometric ratios in terms of its diameter as ... 

Figure 9.5 Fractional occupation, during capture and emission at 350 K, of a pore in porous SiC. An exact calculation is compared with exponential and logarithmic approximations. The filling pulse length is tp, and the boxcar sampling points on the emission transient are t and respectively. Reproduced from [10], with permission from the American Physical Society...

Scientists use mathematical tools and equations to model and solve scientific problems. Solving scientific problems often involves the use of quadratic, trigonometric, exponential, and logarithmic functions. 

In this chapter we have introduced symbolic mathematics, which involves the manipulation of symbols instead of performing numerical operations. We have presented the algebraic tools needed to manipulate expressions containing real scalar variables, real vector variables, and complex scalar variables. We have also introduced ordinary and hyperbolic trigonometric functions, exponentials, and logarithms. A brief introduction to the techniques of problem solving was included. 

In diis section, we will discuss exponential and logarithmic models and their basic characteristics. 

Surface reaction mechanisms include adsorption, desorption, surface nucleation, polynucleation, mononucleation and ion exchange reaction. The dependencies of amounts of precipitate and solution composition on time are different for each mechanism. For example, linear, exponential and logarithmic rate equations are established for volume diffusion, polynuclear growth and spiral growth, respectively. 

Write the exponential and logarithmic relationships of the number 6 and exponent 3 with base 2. 

With exponentials and logarithms, the function might be so rapidly varying that no simple rule is available. In this case, it is best to calculate the smallest and largest values that might occur. 

The continuous distribution functions used in this paper are the well-known exponential and logarithmic normal functions. The exponential function is given by... 

If we fractionate a distribution of negative skewn (Fig. 12), precipitation may obscure this feature completely (see Fig. 37). The set of 6 functions has now been represented by a step function to facilitate comparison with the generalized Beall curves. Then, in particular, representation of the fraction distributions by exponential and logarithmic normal functions (positive skewness) may be very irrelevant. Unfortunately, the fraction data available (weight, M and, at most, M,) do not allow a decision as to the skewn and, haice, there is no way of knowing whether the represraitation used is appropriate. 

The cause of the discrepancies between reconstructed and actual distributions is the insufficient agreement between the fraction distributions and their representation by exponential and logarithmic normal functions. Fig. 11 illustrates this, and an examination of Fig. 29 also shows that actual fraction distributions may readily assume shapes that camiot be accounted for by such simple functions. 

Exponential and Logarithmic Functions. Exponential functions are functions whose defining equation is written in the general form... 

---