18.4 Exponential and Logarithmic Models 603
Another interesting form of an exponential function isf(x) e
x^2. You will find this
type of exponential function in expressing probability distributions. We will discuss probabil-
ity distributions in more detail in Chapter 19. For comparison, we have plotted the functions
f(x) 2
x^2
andf(x) e
x^2
and shown them in Figure 18.14. Note the bell shape of these
functions.
Logarithmic Functions
In this section, we will discuss logarithmic functions. In order to show the importance of
logarithmic functions, we will revisit the cooling of steel plates example, and ask a different
question.The Cooling of Steel Plates (Revisited) In an annealing process, thin steel plates (kthermal
conductivity 40 W/m#K,rdensity 7800 kg /m
3
, andcspecific heat 400 J/ kg#K)
are heated to temperatures of 900 C and then cooled in an environment with temperature of
35 C and a heat transfer coefficient ofh25 W/m(^2) #
K. Each plate has a thickness ofL5 cm.
We are now interested in determining how long it would take for a plate to reach a temperature
TABLE 18.5 Some Important Characteristics of Exponential Functions
Form of the Exponential Function Characteristics
f(x) f 0 a 0
(Example 1)
f(x) f 0 a 0 forf 0 a 0
(Example 2)
e
a 1 x
e
a 1 x
f 0 a 0
f 0
f(x)
x
f 0 a 0
f(x)
x
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