**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.**

**Problem 1:** Find the inverse, if it exists, to the function

If it does not exist, indicate the restricted domain where it will
exist and find the inverse subject to the restricted domain.

**Solution:** From the graph of f(x) you can tell the function is
one-to-one and therefore has an inverse.

We know that the composition of the original function with its
inverse will take us back to x or . Therefore,

All we have to do now is solve for in the equation

**Step 1:**-
can be written
The base is 6 and the exponent is x - 10.

**Step 2:**- Convert the logarithmic equation to an exponential function
with base 6
can be written

**Step 3:**- Isolate by adding 4 to
both side of the equal sign:

Check your solution by graphing both functions and determining whether
they are symmetric the graph of the line y = x.

You can also check
your answer by checking points. In the original equation, the domain
is the set of all real numbers greater than 4, and the range is the
set of all real numbers greater than 10. The domain and range of the
inverse function should be just the reverse.

Choose the x = 40 and calculate f(40). = = 12. This
indicates that (40, 12) is point on the graph of f(x). Let's now see
if the point (12, 40) is a point on the graph of the inverse

The point (12, 40) is on the graph of the inverse.

**If you would like to work another problem, click on Problem.**

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