Logarithm definition

An exact description of the acidity of solutions and correlation of the acidity in various solvents is one of the most important problems in the theory of electrolyte solutions. In 1909, S. P. L. S0rensen suggested the logarithmic definition of acidity for aqueous solutions considering, at that time, of course, hydrogen instead of oxonium ions (cf. Eq. (1.4.11))... 

In other cases, a logarithmic definition is more useful [6, 39] ... 

A plot against Hammett s cr-constants of the logarithms of the rate constants for the solvolysis of a series of Mz-substituted dimethylphenylcarbinyl chlorides, in which compounds direct resonance interaction with the substituent is not possible, yielded a reasonably straight line and gave a value for the reaction constant (p) of — 4 54. Using this value of the reaction constant, and with the data for the rates of solvolysis, a new set of substituent parameters (cr+) was defined. The procedure described above for the definition of cr+, was adopted for... 

Definition of Logarithm. The logarithm x of the number N to the base b is the exponent of the power to which b must be raised to give N. That is,... 

The inherent viscosity (I/C2) In (77/770). A plot of inherent viscosity versus concentration also extrapolates to [77] in the limit of C2 0. That this is the case is readily seen by combining Eq. (9.12) with the definition of the inherent viscosity and then expanding the logarithm ... 

The resolution of this paradox lies in the assumptions about standard (reference), states which are unavoidably involved in the above definitions of and /3l-l- In order to ensure that and /3l-l are dimensionless (as they have to be if their logarithms are to be used) when concentrations are expressed in units which have dimensions, it is necessary to use the ratios of the actual concentrations to the concentrations of... 

Figurel. Equilibrium potential / pH diagram of the Pb/H20 system at 25 °C, according to Pourbaix [10], but simplified for a = lmol L 1. The pH value is used to express the acidity of the solution. Its definition is pH = -log( H.) pH stands for the negative logarithm of the activity of the H+ ions.

There is thus assumed to be a one-to-one correspondence between the most probable distribution and the thermodynamic state. The equilibrium ensemble corresponding to any given thermodynamic state is then used to compute averages over the ensemble of other (not necessarily thermodynamic) properties of the systems represented in the ensemble. The first step in developing this theory is thus a suitable definition of the probability of a distribution in a collection of systems. In classical statistics we are familiar with the fact that the logarithm of the probability of a distribution w[n is — J(n) w n) In w n, and that the classical expression for entropy in the ensemble is20... 

A logarithmic plot of tj versus (fr in Figure 10.13 shows that, using this definition of L, the curves for the slab or platelet and the spherical particle come very close together. 

One of the conditions of spontaneity is that AG < 0 at constant T and P. A new statement of this condition is that in a spontaneous process electrons (or any other substance) move from a state of higher activity to a state of lower activity. Because of the definition of pe as a negative logarithm of activity, the condition of spontaneity for pe is that electrons move from a more negative to a more positive pe. 

Kawabata, Tsuruta, and Furukawa (121) have reported a linear relationship between the logarithms of their Q values and the logarithms of the methyl affinities of Szwarc and co-workers (111, 123, 124). James and MacCallum (125) have found a linear relationship between the logarithms of the Qo values calculated from the definition of Zutty and Burkhart (122) and the logarithms of the rates of addition of ethyl radicals to various substituted ethylenes. Similar... 

A logarithmic scale is useful not only for expressing hydronium ion concentrations, but also for expressing hydroxide ion concentrations and equilibrium constants. That is, the pH definition can be generalized to other quantities pOH = - log [OH ] p Tg = - log Tg p log... 

Use of these definitions and the properties of logarithms leads to a statement of the water equilibrium constant in... 

Series expansion of the natural logarithm demonstrates the equivalence of this definition to Eq. (50). It follows also from Eq. (51) that... 

A problem of all such linear QSPR models is the fact that, by definition, they cannot account for the nonlinear behavior of a property. Therefore, they are much less successful for log S as they are for all kinds of logarithmic partition coefficients. 

A general definition of log P and log D, in its simplest form, can be given as the logarithm of the ratio (P or D) of the concentration of species of interest (the drug in a pharmaceutical context) in each phase, assuming the phases are immiscible and well separated prior to analysis. P is defined as the partition coefficient, whereas D is the distribution coefficient. However, the simplest form does not reveal some of the intricacies of the determination and use of these parameters, and further explanation is necessary. 

Note that the lipophilicity parameter log P is defined as a decimal logarithm. The parabolic equation is only non-linear in the variable log P, but is linear in the coefficients. Hence, it can be solved by multiple linear regression (see Section 10.8). The bilinear equation, however, is non-linear in both the variable P and the coefficients, and can only be solved by means of non-linear regression techniques (see Chapter 11). It is approximately linear with a positive slope (/ ,) for small values of log P, while it is also approximately linear with a negative slope b + b for large values of log P. The term bilinear is used in this context to indicate that the QSAR model can be resolved into two linear relations for small and for large values of P, respectively. This definition differs from the one which has been introduced in the context of principal components analysis in Chapter 17. 

The derivative of the logarithm was already discussed in Chapter 1, while the derivatives of the various trigonometric functions can be developed from their definitions [see, for example, Eqs. (1-36), (1-37), (1-44) and (1-45)]. A number of expressions for the derivatives can be derived from the problems at the end of this chapter. 

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