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 3: Find the inverse of
The base is 10, 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 2 or the set of real numbers greater than 8. 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 .
Recall that the composition
of a function with its inverse will take you back to where you
started. For example, suppose the rule f(x) will take a 3 and
link it to 10; then the rule will take the 10 and
link it back to the 3. Another way of stating this is . A general way to stating this is for any x in the domain of f(x).
which 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 .
We know also that the term
and this implies that
Therefore the inverse is
Check: Let's check our answer by finding points on both graphs. In the original graph
This means that the point (10,1.20411998266) is located on the graph of f(x). If we can show that the point (1.20411998266,10) is located on the inverse, we have shown that our answer is correct, at least for these two points.
indicates that the point (1.20411998266,10) 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.
If you would like to review another example, click on Example.
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Author: Nancy Marcus