# 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.

Example 3: Find the inverse of

Comments: Recall that the composition of a function with its inverse will take you back to where you started. For example, suppose the rule f(x) will take a 3 and link it to 10; then the rule will take the 10 and link it back to the 3. Another way of stating this is . A general way to stating this is for any x in the domain of f(x).

Solution: With respect to this problem,

The base is 10, the exponent is x, and the problem can be converted to the exponential function

If you graph the problem, notice that the graph is not one-to-one. Notice also that the domain is the set of real numbers less than 2 or the set of real numbers greater than 8. To find the inverse of the this function, you will have to restrict the domain to either or .
Suppose that we restrict the domain to the set of real numbers in the interval . Then, the range of the inverse will also be the set of real numbers in the interval .

Recall that the composition of a function with its inverse will take you back to where you started. For example, suppose the rule f(x) will take a 3 and link it to 10; then the rule will take the 10 and link it back to the 3. Another way of stating this is . A general way to stating this is for any x in the domain of f(x).

Step 1:
Solve for in the equation

Step 2:
Subtract 16 from both sides of the equation:

Step 3:
Add 25 to both sides of the above equations:

Step 4:
Factor the right side of the equation in Step 3:

which can be written

Step 5:
Take the square root of both sides:

Step 6:
Add 5 to both sides of the equation:

Step 7:
Now we have a problem because the inverse is supposed to be unique and we have two inverses:

Which one do we choose?
Recall that the range of the inverse equals the domain of the original function. Since we restricted the domain of f(x) to , we know the range of is also .
We know also that the term

and this implies that

Therefore the inverse is

Step 8:
Had we restricted the domain of f(x) to = , the inverse would be

Check: Let's check our answer by finding points on both graphs. In the original graph

This means that the point (10,1.20411998266) is located on the graph of f(x). If we can show that the point (1.20411998266,10) is located on the inverse, we have shown that our answer is correct, at least for these two points.

indicates that the point (1.20411998266,10) 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.

If you would like to review another example, click on Example.

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Author: Nancy Marcus

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