**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 3:** Find the inverse, if it exists, to the function

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

**Solution:** By inspection of the graph of f(x), you can tell the
domain consists of all real numbers to the right of . You know also that you can only take the log of a
positive number, so when you solve 4*x*-7>0, x must be a real number
greater than .

You can tell that the function is one-to-one and therefore has an
inverse. How can you tell from the graph that f(x) is a one-to-one
function? You use the horizontal line test. Run a horizontal line
across the graph; if it intersects the graph more than once at any
given time, it is not one-to-one. Since the inverse exists, you know
that its range is equal to the domain of f(x), or all real numbers
greater than .

You know that

We have to isolate the in the equation

**Step 1:**- Convert the equation to an exponential equation with base 10 by
first adding 8 to both side of the equation:
**Step 2:**- Divide both sides of the above equation by 3:
**Step 3:**- Now convert the above equation to an exponential
equation:
**Step 4:**- Add 7 to both sides of the above equation:
**Step 5:**- Divide both sides by 4:
**Step 6:**- Check your answer by graphing both equations along with
the line y = x. If the graph of f(x) and its inverse are symmetric
to the line y = x, you have calculated the inverse correctly.
**Step 7:**- You can also check it with a few points: Let x = 10.
This means that the point (10, -3.44445818036) is located on the graph of f(x).

If we can show that the point (-3.44445818036, 10) is located on the graph of the inverse, we have illustrated that we have calculated the inverse correctly, at least for these points.

You have correctly worked the problem.

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