Mathematical operations logarithms

Many other mathematical operations are commonly used in analytical chemistry, including powers, roots, and logarithms. Equations for the propagation of uncertainty for some of these functions are shown in Table 4.9. 

You can describe the acidity of an aqueous solution quantitatively by stating the concentration of the hydronium ions that are present. [HsO" ] is often, however, a very small number. The pH scale was devised by a Danish biochemist named Spren Sorensen as a convenient way to represent acidity (and, by extension, basicity). The scale is logarithmic, based on 10. Think of the letter p as a mathematical operation representing -log. The pH of a solution is the exponential power of hydrogen (or hydroni-um) ions, in moles per litre. It can therefore be expressed as follows ... 

Now, how do we solve for P, since it is tied up in the log term Remember that a logarithm is a mathematical operator. To free a quantity from an operator, we need to apply the inverse of that operator. So, if we want to know the value of the pressure, we need to apply the antilog operator to each side of the equation. On your calculator, the antilog button is 10 . So, we find the vapor pressure on enflurane under these conditions is 217 torr. 

For many mathematical operations, including addition, subtraction, multiplication, division, logarithms, exponentials and power relations, there are exact analytical expressions for explicitly propagating input variance and covariance to model predictions of output variance (Bevington, 1969). In analytical variance propagation methods, the mean, variance and covariance matrix of the input distributions are used to determine the mean and variance of the outcome. The following is an example of the exact analytical variance propagation approach. If w is the product of x times y times z, then the equation for the mean or expected value of w, E(w), is ... 

Mathematical functions (arithmetic, logical, relational operators, logarithms, exponents, etc.)... 

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

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

In this chapter, we discuss symbolic mathematical operations, including algebraic operations on real scalar variables, algebraic operations on real vector variables, and algebraic operations on complex scalar variables. We introduce the concept of a mathematical function and discuss trigonometric functions, logarithms and the exponential function. 

The Math Handbook helps you review and sharpen your math skills so you get the most out of understanding how to solve math problems involving chemistry. Reviewing the rules for mathematical operations such as scientific notation, fractions, and logarithms can also help you boost your test scores. 

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... 

In addition to basic arithmetic and algebra, four mathematical operations are used frequently in general chemistry manipulating logarithms, using exponential notation, solving quadratie equations, and graphing data. Each is discussed briefly below. 

Knowing A and for a given reaction in the temperature interval of interest, one can calculate reaction kinetics in nonisothermal experiments. One can also, in reverse, derive values for the activation energy, the pre-exponential factor, and the order of the reaction fi-om nonisothermal experiments. For this purpose one inserts Eq. (7) into Eq. (3) of Fig. 2.8 and gets Eq. (8) as an expression for the nonisothermal reaction kinetics. For analysis, one may take the logarithm on both sides of the equation to make the exponential disappear. In addition, one may differentiate both sides with respect to ln[A], to get an explicit equation for the reaction order n. The result of this mathematical operation is shown in Eq. (9). This is a somewhat arduous equation, usually attributed to Freeman and Carroll. Note that experimentally one knows the parameters concentration, [A], rate, -d[A]/dt,... 

In some cases it is possible to transform the correlation into linear form by mathematical operations, such as taking the logarithm or taking a root. If the reactions are superimposed, it may make sense to consider only the main reactions and ignore other aspects, such as behavior prior to reaching a state of equilibrium [94]. 

Because logarithms are exponents, mathematical operations involving logarithms are similar to those involving exponents as follows ... 

The signal processor is also measurement specific. A different mathematical treatment, such as a logarithmic conversion, is required for data from each kind of sensor, depending on what the operator desires as a readout. Some data treatment is often conducted with computer software. 

In this chapter we have introduced symbolic mathematics, which involves the manipulation of symbols instead of performing numerical operations. We have presented the algebraic tools needed to manipulate expressions containing real scalar variables, real vector variables, and complex scalar variables. We have also introduced ordinary and hyperbolic trigonometric functions, exponentials, and logarithms. A brief introduction to the techniques of problem solving was included. 

Inverse operations are pairs of mathematical manipulations in which one operation undoes the action of the other—for example, addition and subtraction, multiplication and division. The inverse of a number usually means its reciprocal, i.e., x = 1/x. The product of a number and its inverse (reciprocal) equals 1. Raising to a power and extraction of a root are evidently another pair of inverse operations. An alternative inverse operation to raising to a power is taking the logarithm. The following relations are equivalent... 

---