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.
Work through the following examples.
Example 1: Find the inverse of .
The base is 10, the exponent is x, and the problem can be converted to the exponential function which can be simplified to
Recall that the domain of f(x) is equal to the range of
, and the range of f(x) is equal to the domain of . The domain of f(x) is and the range of
is also . In terms of the graphs of these functions, this means that the entire graph of f(x) will be located to the right of the vertical line x = - 1, and the entire graph of
will be located above the line y = - 1.
Let's check our answer by finding points on both graphs. In the original graph . This means that the point (99, 2) is located on the graph of f(x). If we can show that the point (2, 99) is located on the inverse, we have shown that our answer is correct, at least for these two points.
indicates that the point (2, 99) 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.
Example 2: Find the inverse of the function .
can be written
Well Done.
If you would like to review another example,
click on Example.
If you like to work a few problems and check the solutions, click
on Answer below.
Problem 1: Find the inverse, if it exists, to the function
If it does not exist, indicated the restricted domain where it will exist.
Problem 2: Find the inverse, if it exists, to the function
If it does not exist, indicated the restricted domain where it will exist.
Problem 3: Find the inverse, if it exists, to the function
If it does not exist, indicated the restricted domain where it will exist.
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Author: Nancy Marcus