# 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 2: Find the inverse, if it exists, to the function

If it does not exist, indicated the restricted domain where it will exist and find the inverse on the restricted domain..

Solution: You can tell from the graph that the function is not one-to-one. Part of the graph is located to the right of x = 6 and part of the graph is located to the left of x = 2. Therefore, the inverse does not exists on the function's existing domain. We will need to restrict the domain.

can be written

It can be simplified to

for all real numbers greater than 6. Therefore, if we restrict the domain to all real numbers greater than 6, f(x) will have an inverse whose range is all real numbers greater than 6.

provided x is greater than 6. The exponential function will have base 8.

Step 1:
Convert to an exponential equation: This equation can be rewritten as

Step 2:
Subtract 12 from both sides equation:

Step 3:
Add 16 to both sides of the equation:

Step 4:
Factor the right side of the equation:

Step 5:

Step 6:
Add 4 to both sides of the above equation:

Step 7:
We now have a problem. The inverse function must be unique and we have two inverses:

and

Which one do we choose?
We restricted the domain of the original function to the set of real numbers greater than 6. We also know that the range of the inverse is also the set of real numbers greater than 6.
Therefore, is the inverse because

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

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