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

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Название Mathematics for Enzyme Reaction Kinetics and Reactor Performance
Автор произведения F. Xavier Malcata
Жанр Химия
Серия
Издательство Химия
Год выпуска 0
isbn 9781119490333



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= xn = x readily implies

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

2 Graphs illustrating the variation of exponential (left) and logarithm (right) as a function of a real number, displaying a concave up ascending curve and a concave down ascending curve, respectively.

      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)equation

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

      (2.21)equation

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

      (2.22)equation

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

      (2.23)equation

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

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

      – 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,

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

      (2.29)equation

      (2.30)equation

      upon isolation of loga x, one gets

      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