Pharmacology for Anaesthesia and Intensive Care

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6 Mathematics and pharmacokinetics

y = x

y

x

1

1

− 1

− 1

Figure 6.3.The rectangular hyperbola.In general, we are only concerned with the positive

part of this curve. Any function where y is proportional to 1/(x+c), where c is a constant, will

be a rectangular hyperbola.

For any number expressed this way, thebaseis 10 and the logarithm is theexponent.

This describes a logarithmic function, where y=log(x); the scale is such that the

integer values on the y-axis−2,−1, 0, 1, 2, 3,... correspond to 0.01, 0.1, 1, 10, 100,

1000,...onthex-axis – there is no such thing as the logarithm of a negative number

(Figure6.5). If we multiply two numbers, w and z, together we can express each as

an exponent of 10:

w= 10 x and z= 10 y,

so that:

w×z= 10 x× 10 y.

When we multiply two numbers that are expressed as powers of 10 together weadd

their exponents so we can rewrite this multiplication as:

w×z= 10 x× 10 y= 10 (x+y).

Wehave now reduced multiplication to addition and in order to find the result of our

calculation we can convert back using log tables, which was extremely useful in the

days before cheap hand-held calculators were available. What we are adding are the

logarithms of each number, log(w)=x and log(z)=y,so that log(w×z)=x+y.

Forafully expanded example, see the appendix at the end of this chapter.