F. Xavier Malcata - Mathematics for Enzyme Reaction Kinetics and Reactor Performance

with the aid of Eq. (2.10), where straightforward algebraic rearrangement unfolds

(2.13)

that complements Eq. (2.9).

Another essential function is the (natural) exponential, e x– i.e. a power where Neper’s number (ca. 2.718 28) serves as basis; it is sketched in Fig. 2.2a. Note the exclusively positive values of this function – as well as its horizontal asymptote, viz.

(2.14)

The exponential function converts a sum into a product, i.e.

(2.15)

based on the rule of multiplication of powers with the same base; one also realizes that

(2.16)

pertaining to a difference as argument, and obtainable from Eq. (2.15)after replacement of y by −y (since e −yis, by definition, 1/e y). A generalization of Eq. (2.15)reads

(2.17)

where x 1 = x 2 = ⋯ = x n = x readily implies

(2.18)

by virtue of the definition of multiplication as an iterated sum.

Figure 2.2 Variation of (natural) (a) exponential, e x, and (b) logarithm, ln x , as a function of a real number, x .

The inverse of the exponential is the logarithm of the same base, i.e. ln x for the case under scrutiny encompassing e as base; the corresponding plot is labeled as Fig. 2.2b. A vertical asymptote, viz.

(2.19)

is apparent (the concept of limit will be explored in due course); the plot of ln x may be produced from that of e xin Fig. 2.2a, via the rotational procedure referred to above. In terms of properties, one finds that

(2.20)

– so the logarithm converts a product to a sum; in fact, Eq. (2.20)is equivalent to

(2.21)

after taking exponentials of both sides, where Eq. (2.15)supports

(2.22)

– while the definition of inverse function, applied three times, allows one to get

(2.23)

as universal condition, thus guaranteeing validity of Eq. (2.20). If n factors x iare considered, then Eq. (2.20)becomes

(2.24)

should x 1 = x 2 = ⋯ = x n = x hold, then Eq. (2.24)simplifies to

(2.25)

If y is replaced by 1/ y in Eq. (2.20), then one eventually gets

(2.26)

– since ln { x / y } + ln y = ln { xy / y } = ln x as per Eq. (2.20), with isolation of ln { x / y } retrieving the above result; hence, a logarithm transforms a quotient into a difference.

The concept of logarithm extends to bases other than e, say,

(2.27)

a ‐based exponentials may then be taken of both sides to get

(2.28)

– since a ‐based exponential and logarithm are inverse functions of each other. If b ‐based logarithms are taken of both sides, then Eq. (2.28)becomes

(2.29)

in agreement with Eq. (2.25)and after application to Eq. (2.28)– which may, in turn, be combined with Eq. (2.27)to generate

(2.30)

upon isolation of log a x , one gets

(2.31)

Equation (2.31)may be used to convert the logarithm of (any number) x from base b to base a – at the expense of knowledge of log b a ; in particular, one finds that

(2.32)

when a = 10 and b = e, with ln 10 equal to 2.302 59.

The concept of base other than e may indeed be extended to the exponential function itself – and coincides with a plain power, using a as base and the target function as exponent; furthermore, one may state that

(2.33)