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 5: 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_inline45 will take the 10 and link it back to the 3. Another way of stating this is tex2html_wrap_inline47 . A general way to stating this is tex2html_wrap_inline49 for any x in the domain of f(x).

Solution: With respect to this problem,

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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 one-to-one. Therefore, we not have to restrict the domain. Notice that the domain is the set of all real numbers tex2html_wrap_inline57 .

Step 1: Solve for tex2html_wrap_inline59 in the equation

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Step 2: Square both sides:

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Step 3: Add 7 to both sides of the above equations:

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Step 4: Divide both sides by 3:

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

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Author: Nancy Marcus

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