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6 Mathematics and pharmacokinetics20 40 60 80 1002yx-1Figure 6.5.The logarithmic function.This shows the function y=log(x) for logarithms to
the base 10. When x=1y=0,which is a point common to all logarithmic relationships.
Notice that unlike the exponential function, there is no asymptote corresponding to a
maximum value for y. There is an asymptote on the x-axis, since the logarithm approaches
negative infinity as x gets smaller and smaller and approaches 0; there is no such thing as a
logarithm for a negative number.log(10^3 )=log(10× 10 ×10)=log(10)+log(10)+log(10)= 3 ×log(10)=3. So in
our example expression:
log(z)=log(w)+(y×log(10)).
Because log(10)= 1 this simplifies to:
log(z)=log(w)+y.
Sofar we have described the familiar situation where thebaseis 10 and the exponent
is the logarithm to the base 10. We can actually write a number in terms ofany
base, not just 10, with a corresponding exponent as the logarithm to that base. In
pharmacokinetics we are concerned with relationships involving the exponential
function, y=ex,soweuse e as the base for all logarithmic transformations. If we
write a number, say x, in terms of base e, then it is usual to write the logarithm of
xtothe base e as ln(x) rather than log(x) because the latter is often reserved for
logarithms to the base of 10. Logarithms to base e are known asnaturallogarithms;
for example 2 can be written e0.693,sothe natural logarithm of 2, ln(2), is 0.693 (this is
auseful value to remember, as it is the factor that relates time constant to half-life).