Greenwood press

LOGARITHMS
57

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Modeling exponential decay using logarithms (finding half-life)

Further examples on logarithms

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LOGISTIC FUNCTIONS
Logistic functions predict proportions or probabilities. They are used to deter-
mine proportions of successes in “yes–no” situations from underlying factors.
They can be used to predict the proportions of students admitted to a university
from different SAT-score intervals; the probability of getting an item right on a
test depending on underlying knowledge; the probability that a patient with cer-
tain symptoms will die or live; the proportions of nerves in the brain that will fire
in the presence of different concentrations of stimulating chemicals; the spread
of rumors; and the proportion of consumers that will switch brands or stay with
their current one when presented with different saturations of advertising.
A logistic function takes the form 
y =
1
1
m
+b
0
b
x
1
, where m is the maximum 
value of the dependent variable (in most cases, this will be 1.00). The values b
0
and b
1
are very similar to the numbers used in exponential growth models. The
illustration below shows the shape of a logistic function. The scatterplot in it
shows the percent of applications for admission to a large state university that
resulted in acceptances of the candidates. The groupings of students on the x-axis
are by SAT verbal score. The dot at 700 indicates that 95 percent of the appli-
cants who had SAT verbal scores at 700 (that is, in the range of 680–720) were
accepted. However, only 9 percent of the students at 400 (in the range of 380–
420) were accepted. The equation for the logistic curve that models the data is
A =
1
1+9128(0.983)
SAT
, where A is the proportion accepted at an SAT score level.