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

displaymath110

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_inline112 will take the 10 and link it back to the 3. Another way of stating this is tex2html_wrap_inline114 . A general way to stating this is tex2html_wrap_inline116 for any x in the domain of f(x).

Solution: With respect to this problem,

displaymath120

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

displaymath122

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 tex2html_wrap_inline124 or tex2html_wrap_inline126 .
Suppose that we restrict the domain to the set of real numbers in the interval tex2html_wrap_inline126 . Then, the range of the inverse will also be the set of real numbers in the interval tex2html_wrap_inline126 .

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_inline112 will take the 10 and link it back to the 3. Another way of stating this is tex2html_wrap_inline114 . A general way to stating this is tex2html_wrap_inline116 for any x in the domain of f(x).

Step 1:
Solve for tex2html_wrap_inline140 in the equation

displaymath122

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

displaymath144

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

displaymath146

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

displaymath148

which can be written

displaymath150

Step 5:
Take the square root of both sides:

displaymath152

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

displaymath154

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

displaymath156

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_inline126 , we know the range of tex2html_wrap_inline140 is also tex2html_wrap_inline126 .
We know also that the term

displaymath164

and this implies that

displaymath166

Therefore the inverse is

displaymath168

Step 8:
Had we restricted the domain of f(x) to = tex2html_wrap_inline124 , the inverse would be

displaymath172

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

displaymath174

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.

displaymath176

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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