Base of logarithms

If the base of logarithms equals 10, the logarithms are called common logarithms If 10 = y, then X is the common logarithm of y, denoted by logiQ(y). The subscript 10 is sometimes omitted, but this can cause confusion. 

Besides 10, there is another commonly used base of logarithms. This is a transcendental irrational number called e and equal to 2.7182818... 

Generally, biodegrability of the compounds decrease with a drop of water solubility and with an increase of molecular weight and complexity of bond structure (Devinny et al. 1999). The microbial removal of the compounds is limited by its hydrophobicity (lipophilicity), determining the water solubility and thereby pollutant transfer from the gas to the aqueous phase. The hydrophobicity is characterized by the octanol-water partition coefficients. It is a ratio of concentrations of the un-ionized compound in a mixture of two immiscible phases (octanol and water) at equilibrium. It determines the difference in solubility of the compound in these two phases. Estimation of hydrophobicity of the compounds is assessed on the base of logarithm of the ratio (logP). It is assumed, that the compound is hydrophilic... 

Base of logarithms, 24 Boundary conditions, 92, 139 Boyle s law, graph of, 3... 

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,... 

Systems of Logarithms. There are two common systems of logarithms in use (1) the natural (Napierian or hyperbolic) system which uses the base e = 2.71828. . . (2) the common (Briggsian) system which uses the base 10. 

Here, t is the time taken for to fall to 1 /e of its initial value (where e is the base of natural logarithms) and is referred to as the lifetime of state n. If spontaneous emission is the only process by which M decays, comparison with Equation (2.9) shows that... 

Dijferential Operations The following differential operations are valid /, g, are differentiable functions of x, c is a constant e is the base of the natural logarithms. 

AP(j = dry bed pressure drop, in. water/ft AP = operating pressure drop, in. liquid/ft e = base of natural logarithms Xi,X2 = curve fit coefficients for C2, Table 9-32. 

That is, the natural logarithm of a number is 2.303.. . times its base-10 logarithm. 

The base-10 logarithm of an exponential number can be found in a somewhat different way by applying the relation... 

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. 

Logarithm The exponent that indicates the power to which a number is raised to produce a given number. Thus, as an example, 1000 to the base of 10 is 3. This type of mathematics is used extensively in computer software. 

In this expression, Ka is the acidity constant of a weak acid and Kh is the basicity constant of the conjugate base of that acid. The acid and base must form a conjugate acid-base pair (such as CH C00H/CH3C02 or NH4+/NH3). We can express Eq. 1 la in another way by taking logarithms of both sides of the equation ... 

Here B is the world average burden of anthropogenic sulfate aerosol in a column of air, in grams per square meter. The optical depth is then used in the Beer Law (which describes the transmission of light through the entire vertical column of the atmosphere). The law yields I/Iq = where I is the intensity of transmitted radiation, Iq is the incident intensity outside the atmosphere and e is the base of natural logarithms. In the simplest case, where the optical depth is much less than 1, (5 is the fraction of light lost from the solar beam because of... 

The transformation by log column-centering consists of taking logarithms followed by column-centering. The choice of the base of the logarithms has no effect on the interpretation of the result, but decimal logs will be used throughout. 

Consequent to substitution of the value of the standard electrode potentials as obtained from entries in Table 6.11 and conversion of natural logarithm to the base 10 logarithm, the following relationship is obtained ... 

---