Exponents and logarithms

Appendixes Tables of solubility products, acid dissociation constants (updated to 2001 values), redox potentials, and formation constants appear at the back of the book. You will also find discussions of logarithms and exponents, equations of a straight line, propagation of error, balancing redox equations, normality, and analytical standards. 

Significant figures in logarithms and exponents were discussed in Section 3.2. 

Keeping the composition of copolymerization media constant the total comonomer concentration of which is varied. The absorbed dose was kept constant at 0.14 KGy for the AM-AANa and at 0.35 KGy for the AM-DAEA-HCl systems. The results are shown in Figs. 4 and 5, which show the rate of polymerization, Rp, the degree of polymerization, and the intrinsic viscosity increase with increasing monomer concentration. At comonomer concentration >2.1 M/L, DPn decreases with increasing comonomer concentration. From the logarithmic plots, exponents of the comonomer concentration for the AM-AANa system were determined to be [17,54]. 

Because logarithms are exponents, we have the following logarithm laws that are derived from the laws of exponents given on page 8. Let A and B be any two numbers. 

Because logarithms are exponents, they have an intimate connection with exponential functions and with the laws of exponents. 

Logarithms, or logs, can be to different bases. log2 denotes a log to the base 2, and log10 denotes a log to the base 10. Logarithms are exponents. When you solve a log, you are actually calculating the exponent that the base was raised to. 

Figure 1. Logarithms of the ratios of the concentrations to the odor threshold values (log(C/T)) (top) and exponents (n) of the FD factor 2", (bottom) for the isothiocyanates versus their retention indices (RI).

REVIEW OF MATHEMATICAL OPERATIONS EXPONENTS, LOGARITHMS, AND THE QUADRATIC FORMULA... 

There are a number of identities involving logarithms, some of which come from the exponent identities in Eqs. (1.6)-(1.8). Table 1.1 lists some identities involving exponents and logarithms. These identities hold for common logarithms and natural logarithms as well for logarithms to any other base. 

This textbook uses common logarithms based on a base of 10. Therefore, the common log of any number is the power to which 10 is raised to equal that number. Examine Table 5 to compare logs and exponents. Note the log of each number is the power of 10 for the exponent of that number. For example, the common log of 100 is 2, and the common log of 0.01 is —2. 

Because logarithms are exponents, mathematical operations involving logarithms follow the rules for the use of exponents. For example, the product of z" and (where z is any number) is given by... 

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

Because logarithms are exponents, logarithms of products are added and logarithms of quotients are subtracted ... 

The constant a is called the base of the logarithm and the exponent y is called the logarithm ofx to the base a and is denoted by... 

If one plots the macroemulsion lifetime versus spontaneous curvature, one can evaluate the spontaneous curvature coefficient of activation energy, = 35.7/c A/(jo A from the slope of the line and the pre-exponent fo from the intercept. This evaluation resides on the approximations that (i) the pre-exponent itself does not depend on the spontaneous curvature and (ii) the initial droplet size do is close in all the experiments and (iii) changes in T are minor. Of these factors, only (iii) is completely negligible. Fortunately, the contributions from (i) and (ii) are under a logarithm and cannot influence the value of significantly however, they do affect the pre-exponent. 

In the previous examples we did not specify any particular value for the base a that is, the above rules hold for any value of a. In numerical calculations, however, we find that it is convenient to use logarithms to the base 10, since they are directly related to our decimal system of expressing numbers and also are linked to what normally we refer to as scientific notation, in which we express numbers in terms of powers of 10 (e.g., 6.022 x 10 ). Such logarithms are called common logarithms, and are written simply as log y. The relationship between exponents of the number 10 and common logarithms can be seen in Table 3-1. 

Usually, diffusivity and kinematic viscosity are given properties of the feed. Geometiy in an experiment is fixed, thus d and averaged I are constant. Even if values vary somewhat, their presence in the equations as factors with fractional exponents dampens their numerical change. For a continuous steady-state experiment, and even for a batch experiment over a short time, a very useful equation comes from taking the logarithm of either Eq. (22-86) or (22-89) then the partial derivative ... 

Treating k2/(k2 - kj) as an exponent and removing the natural logarithm gives... 

Organic Polymers, Natural and Synthetic 610 Appendix 1 Units, Constants, and Reference Data 635 Appendix 2 Properties of the Elements 641 Appendix 3 Exponents and Logarithms 643 Appendix 4 Nomenclature of Complex Ions 648 Appendix 5 Molecular Orbitals 650... 

Fig. 6-4. Calculated curves showing relationship between intensity ratio and thickness for various values of exponent a. The abscissa scale is logarithmic. Circles = plated coatings squares = evaporated coatings. (Liebhafsky and Zemany, Anal. Chem., 28, 455.)...

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