Logarithms to base ten

The pK value of an acid (pK ) or base (pK,) is defined as the negative logarithm to base ten of the equilibrium dissociation constant K, whereby the more largely positive the pK value the more weak, or less pronounced, the dissociation of the acid (R" H ) or base (R OH ). 

All toxicity values reported here are the negative logarithms to base ten ( p values) of the millimolar concentrations at which a 50% light reduction (Gamma = 1) was observed on 30 min exposure. Each value is the mean of at... 

The bel is used to express values of such logarithmic quantities as field level, power level, sound-pressure level, and attenuation. Logarithms to base ten are used to obtain the numerical values of quantities expressed in bels. The submultiple decibel, dB, is commonly used. 

In the Nernst equation the term RT/nF involves known constants, and introducing the factor for converting natural logarithms to logarithms to base 10, the term has a value at a temperature of 25 °C of 0.0591 V when n is equal to 1. Hence, for an ion M+, a ten-fold change in ionic activity will alter the electrode potential by about 60 millivolts, whilst for an ion M2 +, a similar change in activity will alter the electrode potential by approximately 30 millivolts, and it follows that to achieve an accuracy of 1 per cent in the value determined for the ionic concentration by direct potentiometry, the electrode potential must be capable of measurement to within 0.26 mV for the ion M+, and to within 0.13 mV for the ion M2 +. ... 

Many calculations using Equation refer to standard temperature, 298.15 K. Furthermore,. eq often has a very large or very small value that is expressed using power-of-ten notation. For such values, calculations using the logarithm to base 10 (log) rather than In are more convenient log X-2.302 585 In X. We can substitute these values and the value for F and then evaluate the multiplier for the log term at standard temperature ... 

If we invert the logarithmic quotient to remove the negative sign on the right-hand side, and convert for convenience from natural to base ten logarithms, we may reexpress this relationship as follows ... 

Figure 8.6 Determination of the values of t1/2 and ke from logarithmic plots of plasma concentration against time. The logarithm plot for logs to base ten is shown but the natural logarithmic plot would be similar except the slope would now be equal to kA...

When we convert from natural (In) to base ten logarithms (log) by multiplying by 2.303, the previous equation can be written as... 

Ten is not the only base for logarithms. Many natural phenomena, both chemical and otherwise, involve logarithms to the base e, which is 2.718. Logarithms to base e are known as natural l< arithm. Their value is 2.303 times greater than a base-10 logarithm. Physical chemistry relationships that appear in base e are often converted to base 10 by the 2.303 factor, although modern calculators make it just as easy to work in base e as in base 10. The p concept, however, uses base 10 by definition. 

Values of are small for weak acids and they range very widely (Table 4.1). It is common practice to quote values as the negative logarithm to the base ten, i.e. — logjo K.. since such numbers are less cumbersome and positive when Aj < 1. The symbol for -logio is by convention "p/ fhus -logjo becomes pK,. Table 4.1 shows some typical pAg values. 

Zetmer-logarithmus, m. logarithm to the base 10, common logarithm, -potenz, /. power of ten tenth power, -stein, m. hollow concrete block, -stelle, /. decimal place, -system, n. 

Between these two acids, there is up to a million-fold difference in the number of solvated protons per litre. We cannot cope with the unwieldy magnitude of this difference and tend to talk instead in terms of the logarithm of the concentration. To this end, we introduce a new concept the pH. This is defined mathematically as minus the logarithm (to the base ten) of the hydrogen ion concentration ... 

An acid s pH is defined as minus the logarithm (to the base ten) of the hydrogen ion concentration. 

Since not all electronic calculators are alike, detailed instructions cannot be given here. Read your instruction manual. You should purchase a calculator which, in addition to , —, x, and a functions, provides at least the following scientific notation (powers of ten) logarithms and antilogarithins (inverse logarithms) both natural and common (base ten) and exponentials (y ). If it has these functions, it will probably have reciprocals (1/jt), squares, square roots, and trigonometric functions as well. 

From a practical point of view it is often more convenient to use weight percent for expressing the concentrations in conjunction with the 1 wt% standard state and logarithms to the base ten. 

When you perform calculations, such as using half-life of carbon to determine the age of the skull in Figure 22 or the pH of the products in Figure 23, you may need to use the log or antilog function on your calculator. A logarithm (log) is the power or exponent to which a number, called a base, must be raised in order to obtain a given positive number. This textbook uses common logarithms based on a base of 10. Therefore, the common log of any number is the power to which ten is raised to equal that number. Examine Table 4. Note the log of each number is the power of ten for the exponent of that number. For example, the common log of 100 is two and the common log of 0.01 is -2. 

Now take the negative logarithm to the base ten of each side ... 

The final relation uses a derivative with respect to the base ten logarithm of molar mass, as is customary for the calibration curve (see Fig. 1.26). The number density distribution function n p,N) is related to vv f in a... 

It is not necessary to speak of negative logarithms to the base ten . It is more important to understand that the dilution of an acid by 1 10 results in an increase of the pH value by 1 (see E7.12). That also means, that the pH value 4 of mineral water correlates to a concentration of H30 + (aq) ions of 10-4 mol/1 the mineral water therefore shows a minimal acidic effect. If one takes the 10-1 molar hydrochloric acid solution in the laboratory and wants to have the same pH 4 solution, one has to dilute this hydrochloric acid to the factor of 1 1000 (see E7.12). Should the pH value 6 of rain water be simulated (rain water is slightly acidic through the reaction of water with carbon dioxide in the air), then 1 ml 10 1 molar hydrochloric acid has to be further diluted with distilled water to 100 1 (volume of a bathtub). 

Capon and Overend pointed out that it had been customary for carbohydrate chemists (especially prior to 1950) to use common logarithms (to the base ten) instead of natural logarithms in determining rate constants. As in their Chapter, all rate constants in this Chapter have been converted into the values based on natural logarithms, with the second as the unit of time. However, the integrated rate expression used in the original articles was not always apparent, and note has been made of this when there is doubt. 

For logarithmic and all other functions, we do the same as we have done with Sin. It is important to know, however, that the function Log in Mathematica is the natural logarithm and not the log base ten. 

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