If the logarithmic function is one-to-one,
its inverse exits. The inverse of a logarithmic function is
an exponential function. When you graph both the logarithmic
function and its inverse, and you also graph the line y = x,
you will note that the graphs of the logarithmic function and
the exponential function are mirror images of one another with
respect to the line y = x. If you were to fold the graph along
the line y = x and hold the paper up to a light, you would note
that the two graphs are superimposed on one another. Another
way of saying this is that a logarithmic function and its inverse
are symmetrical with respect to the line y = x.
Example 4: Find the inverse of
The base is e, the exponent is x, and the problem can be converted to the exponential function
If you graph the problem, notice that the graph is not one-to-one.
Notice also that the domain is the set of real numbers less than 3
or the set of real numbers greater than 3. To find the inverse of the
this function, you will have to restrict the domain to either or .
Suppose that we restrict the domain to the set of real numbers in the interval . Then, the range of the inverse will also be the set of real numbers in the interval .
can be written
Which one do we choose? Recall that the range of the inverse equals
the domain of the original function. Since we restricted the domain
of f(x) to , we know the range of
is also .
Therefore the inverse is
Let's check our answer by finding points on both graphs. In the
original graph . This
means that the point (5,1.60943791243) is located on the graph of
f(x). If we can show that the point (1.60943791243,5) is located on
the inverse, we have shown that our answer is correct, at least for
these two points. indicates that the point
(1.60943791243,5) is located on the graph of the inverse function. We
have correctly calculated the inverse of the logarithmic function
f(x). This is not the ``pure'' proof that you are correct; however,
it works at an elementary level.
Here is another example.
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Author: Nancy Marcus