Discrete Mathematics Demystified

64 Discrete Mathematics Demystified
Year Amount
After the first year $1000 · 1.05
After the second year $1000 · (1.05) 2
After the third year $1000 · (1.05) 3
···
After the k th year $1000 · (1.05) k
The most important fact about an exponential function is that it will grow faster
than a polynomial function. We can illustrate this fact simply by doing some calculations
with a calculator:
Variable x f(x) = x 2 g(x) = 2 x
1 1 2
2 4 4
3 9 8
4 16 16
10 100 1024
50 2500 1.125 × 10 15
100 10000 1.26 × 10 30
It can be proved in general that if p is any polynomial and f is any exponential
function with positive exponent then, for x large enough (and positive),
f (x) >p(x). In fact we can say more: For x large enough, f (x) >100 · p(x).Or,
for x even larger, f (x) >1000 · p(x). As we see from the last table, the growth of
exponential functions is dramatic.
The flip side of this last information is that logarithmic functions—a third important
type of function—grow very slowly (in fact slower than any polynomial). A
logarithmic function is simply the inverse of an exponential function. For example,
h(x) = log 2 x is the inverse of g(x) = 2 x . We illustrate the idea with another table:
Variable x f(x) = x 2 g(x) = log 2 x
1 1 0
2 4 1
4 16 2
16 256 4
32 1024 5
128 16384 7