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 4: 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 e, 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 3 or the set of real numbers greater than 3. 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 .

Step 1: Solve for

in the equation

Step 2: Subtract 10 from both sides of the equation:

Step 3: Add 9 to both sides of the above equations:

Step 4: Factor the right side of the equation in

can be written

Step 5: Take the square root of both sides:

Step 6: Add 3 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:

and

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 .
Therefore the inverse is

Step 8: Had we restricted the domain of f(x) to =

, the inverse would be

Let's check our answer by finding points on both graphs. In the original graph . This means that the point (5,1.60943791243) is located on the graph of f(x). If we can show that the point (1.60943791243,5) is located on the inverse, we have shown that our answer is correct, at least for these two points. indicates that the point (1.60943791243,5) 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.

Here is another example.

[Back to Rules of Logarithms] [Back to Exponential Functions]

[Algebra] [Trigonometry] [Complex Variables]

S.O.S MATHematics home page