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► Problems for "The Number e and the Natural Log"
Inverse, Exponential, and Logarithmic Functions
The Number e and the Natural Log
The Number e
e is a number that can be defined in many ways. First,
e = 1+
Also, e is the number such that
= 1
The numerical value of e is approximately 2.71828.... The function f (x) = ex is an exponential function, which is a
function of the form f (x) = ax, where a is a positive constant. The graph of f (x) = ex is shown below.
Figure 2.1:Graph of the function f (x) = ex
The Natural Log
The inverse of the function ex is the natural log function ln(x). These two graphs
are pictured below:
Figure 2.2:ex and ln(x) are inverses
Recall that if logab = x, then x is the power that a must be raised to in order to
equal b.
With the natural log, the base is e. So, for example, ln(e) = 1. Because ex and ln(x) are inverses, the following
relations hold:
eln(x) = x and ln(ex) = x
Basic Properties of Logarithms
As a review, recall the following basic rules of logarithms:
1+
= 1
= log(M) - log(N)
