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 1: Find the inverse, if it exists, to the function

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

Solution: From the graph of f(x) you can tell the function is one-to-one and therefore has an inverse.
We know that the composition of the original function with its inverse will take us back to x or tex2html_wrap_inline44 . Therefore,

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All we have to do now is solve for tex2html_wrap_inline48 in the equation

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Step 1:
tex2html_wrap_inline52 can be written

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The base is 6 and the exponent is x - 10.

Step 2:
Convert the logarithmic equation tex2html_wrap_inline56 to an exponential function with base 6

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can be written

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Step 3:
Isolate tex2html_wrap_inline48 by adding 4 to both side of the equal sign:

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Check your solution by graphing both functions and determining whether they are symmetric the graph of the line y = x.
You can also check your answer by checking points. In the original equation, the domain is the set of all real numbers greater than 4, and the range is the set of all real numbers greater than 10. The domain and range of the inverse function should be just the reverse.
Choose the x = 40 and calculate f(40). tex2html_wrap_inline66 = tex2html_wrap_inline68 = 12. This indicates that (40, 12) is point on the graph of f(x). Let's now see if the point (12, 40) is a point on the graph of the inverse

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The point (12, 40) is on the graph of the inverse.

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

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Author: Nancy Marcus

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