PROPERTIES OF LOGARITHMS

SOLVING LOGARITHMIC EQUATIONS

1. To solve a logarithmic equation, rewrite the equation in exponential form and solve for the variable.

Example 5: Consider the function

displaymath98

Find the inverse tex2html_wrap_inline100 .

Solution:

Step 1: From the graph, we determine that the function f(x) is one-to-one and therefore has a unique inverse. How can you tell from the graph that a function is one-to-one? Use the horizontal line test. Mentally run a horizontal line over the graph. If at any time, the line touches the graph in more than one place, it is not a one-to-one function.
Step 2: Since f(x) is a one-to-one function, we know that the inverse exists. The inverse of a logarithmic function is an exponential function.
Step 3: We know that the domain and range of f(x) are equal to the range and domain of tex2html_wrap_inline102 . From the graph of f(x), the domain is the set of all real numbers, and the range is the set of all positive real numbers.
Step 4: We know that the composition of a function with its inverse will yield x or

displaymath104

Step 5: Find f(inverse):

displaymath106

Step 6: Convert the equation to an exponential equation:

displaymath108

Step 7: Isolate the tex2html_wrap_inline100 term using steps 8 through 13.
Step 8: Subtract tex2html_wrap_inline102 from both sides of the above equation:

displaymath114

Step 9: Square both sides of the above equation:

displaymath116

Step 10: Expand the left side of the above equation:

displaymath118

Step 11: Subtract tex2html_wrap_inline120 from both sides of the above equation:

displaymath122

Step 12: Subtract tex2html_wrap_inline124 from both sides of the above equation:

displaymath126

Step 13: Divide both sides of the above equation by tex2html_wrap_inline128 to get

displaymath130

Check: You can check the problem by graphing the function f(x), graphing its inverse, and graphing the line y = x. If the graphs of the function and its inverse are symmetric to the line y = x, you have correctly found the inverse.
You can also check your answer by finding a point on the original graph, say (a, b), and determine whether the point (b,a) is on the graph of the inverse. This is not a perfect check, but it will alert you of a wrong answer.
For example,

displaymath132

indicates that the point tex2html_wrap_inline134 is a point on the graph of f(x).

displaymath136

indicates that the point tex2html_wrap_inline138 is located on the graph of tex2html_wrap_inline100 .

If you would like to review another example, click on Example.

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Author: Nancy Marcus

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