# LOGARITHMS AND THEIR INVERSES

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 4x-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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