Page 161 - Calculus Demystified

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148 CHAPTER 6 = Transcendental Functions
−2
f(−2) = 3
0 9
f(0) = 3 = 1.
The inverse of the function f is the function g which assigns to x the power to
which you need to raise 3 to obtain x. For instance,

g(9) = 2 because f(2) = 9
g(1/27) =−3 because f(−3) = 1/27
g(1) = 0 because f(0) = 1.
We usually call the function g the “logarithm to the base 3” and we write g(x) =
log x. Logarithms to other bases are defined similarly.
3
While this approach to logarithms has heuristic appeal, it has many drawbacks:
we do not really know what 3 means when x is not a rational number; we have
x
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no way to determine the derivative of f or of g; we have no way to determine the
integral of f or of g. Because of these difficulties, we are going to use an entirely
new method for studying logarithms. It turns out to be equivalent to the intuitive
method described above, and leads rapidly to the calculus results that we need.

6.1.1 A NEW APPROACH TO LOGARITHMS
When you studied logarithms in the past you learned the formula

log(x · y) = log x + log y;
this says that logs convert multiplication to addition. It turns out that this property
alone uniquely determines the logarithm function.
Let (x) be a differentiable function with domain the positive real numbers
and whose derivative function (x) is continuous. Assume that satisfies the

multiplicative law
(x · y) = (x) + (y) (∗)
for all positive x and y. Then it must be that (1) = 0 and there is a constant C
such that
C
(x) = x .

In other words
x
(x) = C dt.
1 t
A function (x) that satisfies these properties is called a logarithm function.
The particular logarithm function which satisfies (1) = 1 is called the natural

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