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

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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 tex2html_wrap_inline81 will take the 10 and link it back to the 3. Another way of stating this is tex2html_wrap_inline83 . A general way to stating this is tex2html_wrap_inline85 for any x in the domain of f(x).

Solution: With respect to this problem,

displaymath89

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

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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 tex2html_wrap_inline93 or tex2html_wrap_inline95 .
Suppose that we restrict the domain to the set of real numbers in the interval tex2html_wrap_inline95 . Then, the range of the inverse will also be the set of real numbers in the interval tex2html_wrap_inline95 .

Step 1: Solve for tex2html_wrap_inline101 in the equation

displaymath91

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

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

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Step 4: Factor the right side of the equation in

displaymath107

can be written

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Step 5: Take the square root of both sides:

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Step 6: Add 3 to both sides of the equation:

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

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and

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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 tex2html_wrap_inline95 , we know the range of tex2html_wrap_inline101 is also tex2html_wrap_inline95 .
Therefore the inverse is

displaymath117

Step 8: Had we restricted the domain of f(x) to = tex2html_wrap_inline93 , the inverse would be

displaymath119

Let's check our answer by finding points on both graphs. In the original graph tex2html_wrap_inline133 . 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. tex2html_wrap_inline135 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.

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Author: Nancy Marcus

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