Logarithm

The logarithm (log, in abbreviated form) of a positive number is the exponent, or power, of a given base that is required to produce that number. For example, since 1000 = 10, 100= 10, 10= lO, 1 = 10°, then the logarithms of 1 000, 100, 10, 1, to the base 10 are respectively 3, 2, 1, 0. It is obvious that will give some number greater than 10 (which is lO ) but smaller than 100 (10 ). Actually, 

The system of logarithms whose base is 10 (called the common or Briggsian system) had been widely used for making numerical computations (multiplication and division) before the advent of electronic calculators and computers. The techniques will be presented later in this appendix and numerical tables are provided in Appendix D. Students who have calculators will not need these techniques but should learn them for the sake of improving their understanding of logarithms and as a backup method of calculation. 

The logarithmic function that occurs commonly in physics and chemistry as part of the solution to certain differential equations has as its base not the number 10 but the transcendental number e = 2.718 28. To differentiate between the common and the natural or Napierian logarithms, a more explicit notation could be used logio N = x and log N = y, where 10 = N and e = N. In this book, and in many chemistry and physics books, the notation log N is used to indicate the logarithm to the base 10, and In N to indicate the natural logarithm to the base e. 

Although it is possible to construct logarithm tables for the base e, or for any other base for that matter, there is a simple relationship between any two base systems such that a number found by the use of the common logarithmic table can easily be converted to the logarithm to any other base. Specifically, for the interconversion of common and natural logarithms, it can be shown that 

The logarithm of a number greater than I is positive of a number less than 1 it is negative. The logarithm of 0 is infinitely negative. 

Inverse operations are pairs of mathematical manipulations in which one operation undoes the action of the other—for example, addition and subtraction, multiplication and division. The inverse of a number usually means its reciprocal, i.e., x = 1/x. The product of a number and its inverse (reciprocal) equals 1. Raising to a power and extraction of a root are evidently another pair of inverse operations. An alternative inverse operation to raising to a power is taking the logarithm. The following relations are equivalent 

All the formulas for manipulating logarithms can be obtained from corresponding relations involving raising to powers. If x = a, then x = a . The last relation is equivalent to ny = log fx ) therefore. 

There is no simple reduction for log(x-l-y)—do not fall into the trap mentioned in the preface Since a = a. 

The log of a number less than 1 has a negative value. For any base a 1, = 0, so that 

To find the relationship between logarithms of different bases, suppose x = b, so y = log X. Now, taking logs to the base a. 

For these reasons, and for their own convenience, scientists devised a much simpler method of expressing [H+]. It is called the pH scale. The pH scale expresses the acidity level of a solution in easily understood and manageable numbers. The pH of a solution is represented by the negative logarithm of the [H1. 

The logarithm of a number is the power to which 10 must be raised to give the number. For example, the power to which 10 must be raised to give the number 100 is 2. Thus the logarithm of 100 is 2. Similarly, the power to which 10 must be raised to give the number 1000 is 3. Thus the logarithm of 1000 is 3. See Table 12.4 for other examples. 

All positive numbers have a logarithm, not just those that are multiples of 10. The power to which 10 must be raised in most cases is not a whole number like it is for the multiples of 10 in Table 12.4. For example, the logarithm of 2 is 0.30103. The logarithm of 3 is 0.4771213. The logarithm of 5.11932 is 0.7092123 (Table 12.5). 

It is also important to be able to calculate the [H+] given the pH. Here is an example of this for practice If the pH of a solution is 3.51, what is the [H+] The answer is 3.1 x 10 M. Additional examples follow. 

Sometimes it may be necessary to calculate the pH of a solution when the concentration of an acid is given, but not the [Hi. If the acid is a strong acid, it completely ionizes in water solution, and thus the acid concentration and the [Hi are the same because the number of hydrogen ions present equals the number of acid molecules that are dissolved. 

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T = temperature equivalent at atmospheric pressure T = experimental temperature taken at pressure P P = pressure log = common logarithm (base 10)... 

Watson characterization factor log = common logarithm (base 10)... 

Once the production potential of the producing wells is insufficient to maintain the plateau rate, the decline periodbegins. For an individual well in depletion drive, this commences as soon as production starts, and a plateau for the field can only be maintained by drilling more wells. Well performance during the decline period can be estimated by decline curve analysis which assumes that the decline can be described by a mathematical formula. Examples of this would be to assume an exponential decline with 10% decline per annum, or a straight line relationship between the cumulative oil production and the logarithm of the water cut. These assumptions become more robust when based on a fit to measured production data. 

In the simplest case, for a pressure drawdown survey, the radial inflow equation indicates that the bottom hole flowing pressure is proportional to the logarithm of time. From the straight line plot ot pressure against the log (time), the reservoir permeability can be determined, and subsequently the total skin of the well. For a build-up survey, a similar plot (the so-called Horner plot) may be used to determine the same parameters, whose values act as an independent quality check on those derived from the drawdown survey. 

Figure 3. Brittle material AE responses as count velocity N and logarithm spectrum log (S) characteristics of the process...

K) is the Fourier transform of the logarithmic fluctuation of acoustics impedance. 

The ultrasound system should have more independent channels and allow the transmitter pulse to be individually adjustable in width and amplitude, and an increased frequency range for the logarithmic amplifier was desired. The digitization should be improved both with respect to sampling rate and resolution. 

LOG Logarithmic amplifier with 60/100 dB dynamic range without gain setting 0.1 to 10 MHz (-3 dB)... 

The HILL-SCAN 3020LOG with a logarithmic amplifier provides A-seans with a single-shot dynamic range of 100 dB. 

Because of the double sound path involved in PE measurements of the back wall echo, we approximate the corresponding attenuation at a certain frequency to be twice as large as the attenuation that would be obtained by an ordinary TT measurement. We propose to use the logarithm of the absolute value of the Fourier transform of the back wall echo as input data, i.e... 

In Figure 3 we see how the logarithm of the spectral amplitude effects the estimation results. For each component in input data vector, u, we have defined the feature relevance, Fn d), as... 

Figure 3 Feature relevance. The weight parameters for every component in the input vector multiplied with the standard deviation for that component are plotted. This is a measure of the significance of this feature (in this case, the logarithm of the power in a small frequency region.)...

The kinetics of reactions in which a new phase is formed may be complicated by the interference of that phase with the ease of access of the reactants to each other. This is the situation in corrosion and tarnishing reactions. Thus in the corrosion of a metal by oxygen the increasingly thick coating of oxide that builds up may offer more and more impedance to the reaction. Typical rate expressions are the logarithmic law,... 

We have seen various kinds of explanations of why may vary with 6. The subject may, in a sense, be bypassed and an energy distribution function obtained much as in Section XVII-14A. In doing this, Cerefolini and Re [149] used a rate law in which the amount desorbed is linear in the logarithm of time (the Elovich equation). 

The Debye-Htickel limiting law predicts a square-root dependence on the ionic strength/= MTLcz of the logarithm of the mean activity coefficient (log y ), tire heat of dilution (E /VI) and the excess volume it is considered to be an exact expression for the behaviour of an electrolyte at infinite dilution. Some experimental results for the activity coefficients and heats of dilution are shown in figure A2.3.11 for aqueous solutions of NaCl and ZnSO at 25°C the results are typical of the observations for 1-1 (e.g.NaCl) and 2-2 (e.g. ZnSO ) aqueous electrolyte solutions at this temperature. 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

Figure Bl.15.3. Typical magnitudes of interactions of electron and nuclear spins in the solid state (logarithmic scale).

The concentration at which micellization commences is called the critical micelle concentration, erne. Any experimental teclmique sensitive to a solution property modified by micellization or sensitive to some probe (molecule or ion) property modified by micellization is generally adequate to quantitatively estimate the onset of micellization. The detennination of erne is usually done by plotting the experimentally measured property or response as a hmction of the logarithm of the surfactant concentration. The intersection of asymptotes fitted to the experimental data or as a breakpoint in the experimental data denotes the erne. A partial listing of experimental... 

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