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.

Problem 3: Find the inverse, if it exists, to the function

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If it does not exist, indicate the restrict domain where it will exist and find the inverse over the restricted domain..

Solution: By inspection of the graph of f(x), you can tell the domain consists of all real numbers to the right of tex2html_wrap_inline59 . You know also that you can only take the log of a positive number, so when you solve 4x-7>0, x must be a real number greater than tex2html_wrap_inline59 .
You can tell that the function is one-to-one and therefore has an inverse. How can you tell from the graph that f(x) is a one-to-one function? You use the horizontal line test. Run a horizontal line across the graph; if it intersects the graph more than once at any given time, it is not one-to-one. Since the inverse exists, you know that its range is equal to the domain of f(x), or all real numbers greater than tex2html_wrap_inline59 .
You know that

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We have to isolate the tex2html_wrap_inline69 in the equation

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Step 1:
Convert the equation tex2html_wrap_inline73 to an exponential equation with base 10 by first adding 8 to both side of the equation:

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Step 2:
Divide both sides of the above equation by 3:

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Step 3:
Now convert the above equation to an exponential equation:

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Step 4:
Add 7 to both sides of the above equation:

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Step 5:
Divide both sides by 4:

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Step 6:
Check your answer by graphing both equations along with the line y = x. If the graph of f(x) and its inverse are symmetric to the line y = x, you have calculated the inverse correctly.

Step 7:
You can also check it with a few points: Let x = 10.

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This means that the point (10, -3.44445818036) is located on the graph of f(x).
If we can show that the point (-3.44445818036, 10) is located on the graph of the inverse, we have illustrated that we have calculated the inverse correctly, at least for these points.

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You have correctly worked the problem.

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Author: Nancy Marcus

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