Work with logarithms

Logarithms (or more usually logs) were invented to make manipulation of awkward numbers easy, but calculators have made it even easier. We can use logs as a tool without a discussion of the theory behind them, but a simple explanation is given at the end of the unit. 

Logs are easy on a calculator. You simply press the log key for logic. (Some calculators require that you press the log button before the number and the equals button after the number, others just require the number followed by the log button practise with your own calculator.) 

Inverse logs (or anti-logs) convert a log back into a normal number, and usually involve using two keys, a shift or inverse or mode key and the appropriate log key. 

Practise will tell you which key is required for your particular calculator. 

Convert this answer back to its original form using the inverse log key sequence. If you don t get 6.1 X 10 then you have gone wrong somewhere. 

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Note To find e-95 66, take the inverse In of-95.66 on your calculator, inv In of-95.66 = 2.85 x 10 42. Keep one more significant figure and round off to three significant figures at the end, particularly when working with logarithms. 

Note Keep all the significant figures and round at the end. Remember the number of decimal places in pH or pOH values are set by the number of significant figures in the [H+] or [OH-] this is a result of working with logarithms. 

It is convenient in mathematical operations, even when working with logarithms, to express numbers semiexponentially. Mathematical operations with exponents are summarized as follows ... 

Analysts and researchers frequendy work with logarithms of yields and prices, or continuously compounded rates. One advantage of the logarithmic approach is that it converts the nonlinear relationship expressed in (3-2) into a linear one. The zero-coupon bond price equation in continuous time is... 

Since it is more convenient to work with logarithms to the base 10, the value of the coefficient RT/F is multiplied by the conversion factor 2.303. Then, from the value of i = 8.314 J/deg-mole, T = 298.2K, and F= 96,500C/eq, the coefficient 2303 RT/F at 25 °C becomes 0.0592 V. This coefficient appears frequently in expressions representing potentials or emf. 

These properties are very useful in working with logarithms. 

Addition of dedbels Since decibels are logarithmic units, they are not added arithmetically. Decibels are added by converting them to power, intensity, or pressnre adding these quantities arithmetically and then converting them back to decibels. The mathematics involved may be somewhat complex for people not accustomed to working with logarithms, so charts have been developed for convenience. See Fig. 4.97 for one example of such a chart. To add the sound levels of 80 dB and 74 dB that are produced individually by... 

In the remaining sections of this chapter you will be multiplying and dividing exponentials, taking the square root of an exponential, and working with logarithms. We will furnish brief comments on these operations as we come to them. For more detailed instructions, see Appendix I, Parts B and C. 

Every chemistry student uses a calculator to solve chemistry problems. A suitable calculator can (1) add, subtract, multiply, and divide (2) perform these operations in exponential notation (3) work with logarithms and (4) raise any base to a power. Calculators that can perform these operations usually have other capabilities, too, such as finding squares and square roots, carrying out trigonometric functions, and offering shortcuts for pi and percentage, enclosures, statistical features, and different levels of storage and recall. 

Because logarithms are exponents, the rules governing the use of exponents apply as well The rules that follow are valid for all types of logarithms, regardless of the base. We illustrate the rules with natural logarithms that is where you are most likely to use them in working with this text. 

In practice, most electrochemists work with logig. If written in terms of logig, the two large brackets on the right-hand side of this logarithmic version of equation (7.16) should be multiplied by a factor of 2.303 . 

We assume you have a basic facility with algebra and arithmetic. You should know how to solve simple equations for an unknown variable. You should know how to work with exponents and logarithms. That s about it for the math. At no point do we ask you to, say, consider the contradictions between the Schrodinger equation and stochastic wavefunction collapse. 

Solving a logarithmic equation In working with the Nemst and Henderson-Hasselbalch equations, we will need to solve equations such as... 

Working with the properties of logarithms and trigonometric identities. 

When the deviations between the two distributions are small, the logarithm becomes to leading order the relative deviation ntot(a)/n ((r) - 1 and we can identify S1 2 as the root-mean-square relative deviation. We work with the logarithm rather than directly with the relative deviation because the former gives more sensible behavior for larger deviations in particular, isolated points where ntot(cr) is nonzero while n (cr) is close to zero do not lead to divergences of <5 [54],... 

For k = 0, —00, +00, indeterminate forms appear, most easily resolved by working with the logarithm ... 

Become familiar with a few properties of logarithms. There are several formulas that require the use of logarithms. Because logarithms are easy to work with on a calculator, you may never have learned much about them. Becoming familiar with a few properties of logarithms can help you work more quickly on some problems, especially pH problems. 

We can maximize the number W by using the method of Lagrange multipliers. Again, it is convenient to work with the logarithm of the number W, which allows... 

Spreadsheets have program-specific sets of predetermined functions but they almost all include trigonometrical functions, angle functions, logarithms (p. 262) and random number functions. Functions are invaluable for transforming sets of data rapidly and can be used in formulae required for more complex analyses. Spreadsheets work with an order of preference of the operators in much the same way as a standard calculator and this must always be taken into account when operators are used in formulae. They also require a very precise syntax - the program should warn you if you break this ... 

It Analyze heating and cooling of a fluid flowing in a tube under constant surface temperature and constant surface heat flux conditions, and work with the logarithmic mean temperature difference,... 

Of special interest for the topic of the present chapter is the observation of Weaver that while the double-layer-corrected AS quantities are ligand sensitive, they are found to be independent of potential. This is not the case for the atom and electron transfer process involved in the hydrogen evolution reaction at Hg studied by Conway, et where an appreciable potential dependence of AS is observed, corresponding to conventionally anomalous variation of the Tafel slope with temperature. Unfortunately, in the work with the ionic redox reactions, as studied by Weaver, it is only possible to evaluate the variation of the transfer coefficient or symmetry factor with temperature with a limited variety of redox pairs since Tafel slopes, corresponding to any appreciable logarithmic range of current densities, are not always easily measurable. Alternatively, evaluation of a or /3 from reaction-order determination requires detailed double-layer studies over a range of temperatures. 

PH Value. An operational use of the pH term should occure naturally after students have tested various household solutions with universal indicator paper with pH scale - they don t have to deal with any logarithm (see Fig. 7.9). It is more difficult to teach the quantitative meaning of pH value one has to work with concentrations, the logarithm and the mol term. In this case, it is advantageous to relate the meaning of 1 mol to a specific amount of small particles and to decide the type of particles, which are to be counted 18 g water do not contain 1 mol of water , but rather 1 mol of H20 molecules . A liter of 1 M hydrochloric acid contains 1 mol of H30 + (aq) ions and 1 mol of Cl (aq) ions the concentration is equal to 1 mol/1 for both kinds of ions. Dilution in the volume ratio 1 10 results in a solution with the H + (aq) ion concentration of 0.1 mol/1, the dilution 1 100 leads to the 0.01 M or 10 2 M solution. 

Because the concentrations of and OH ions in aqueous solutions are frequently very small numbers and therefore inconvenient to work with, Soren Sorensen in 1909 proposed a more practical measure called pH. The pH of a solution is defined as the negative logarithm of the hydrogen ion concentration (in mol/L) ... 

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