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Section IBasic principlesWemanipulate natural logarithms in the same way as we do logarithms to the base- In our pharmacokinetic models we come across the expression C=C 0 e−ktand
wecan convert this to its logarithmic equivalent. Because we have an expression
involving e it makes sense to use natural logarithms rather than logarithms to the
base 10. Using the same arguments as given earlier for logarithms to the base 10 we
can rewrite this expression as:
ln(C)=ln(C 0 e−kt)
ln(C)=ln(C 0 )+ln(e−kt).
Inthe same way that log(10^3 )= 3 ×log(10), ln(e−kt) can be written as−kt×ln(e),
so:
ln(C)=ln(C 0 )+(−kt×ln(e)).
For the same reason that log(10)=1, ln(e)=1 (because e=e^1 ) and:
ln(C)=ln(C 0 )−kt.
This gives the equation of the straight line we met above, with the gradient−k and
ln(C 0 ) the intercept on the ln(C) axis. Thus for a simple exponential relationship only
two pieces of information are needed to draw both this line and the exponential it
has been derived from, namely the rate constant for elimination and the intercept
on the concentration axis, C 0 (Figure6.6).
There is a simple relationship between logarithms to the base 10 and natural
logarithms. For plasma concentration, C, we can write:
C= 10 log(C)=eln(C).
Taking natural logarithms of both sides this can then be written:
ln(C)=ln(10log(C))
=log(C)×ln(10).
Equally we could take log to the base 10 of each side and find that:
log(C)=log(eln(C))
=ln(C)×log(e).
So,ifweknow the natural logarithm of a number we can find its logarithm to base 10
simply by dividing by the natural logarithm of 10 (2.302); if we know the logarithm
to base 10 we find its natural logarithm by dividing by the logarithm to the base 10
of e (0.434).
Inpharmacokinetics we do a semi-log plot of concentration against time. It is
easier to use natural logarithms for the concentration axis, because this gives a slope
of−k for the resultant straight line (ln(C)=ln(C 0 )−kt). If logarithms to the base
10 are taken the equation of the straight line is log(C)=log(C 0 )−k log(e)t so the
slope will be different, by a factor of log(e). The other relationship that requires a
natural logarithm as a factor is the relationship between time constant and half-
life. Time constant (τ)isthe inverse of the rate constant; it is the time taken for the