## SOLVING LOGARITHMIC EQUATIONS

Note:

If you would like an in-depth review of logarithms, the rules of logarithms, logarithmic functions and logarithmic equations, click on logarithmic function

Solve for x in the following equation.

Example 4:

Note that the domain of is the set of real numbers such that or when because you cannot take the log of zero or a negative number. Note that the expression is always positive except at

Isolate the logarithmic term.

The exact answers are and the approximate answers are

Check the answer by substituting in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

• Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 12.5427684616 for x, then is a solution.

Check the answer by substituting in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

• Left Side:

• Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value -7.54276846159 for x, then is a solution.

You can also check your answer by graphing (formed by subtracting the right side of the original equation from the left side). Look to see where the graph crosses the x-axis; that will be the real solution. Note that the graph crosses the x-axis at 12.5427684616 and -7.54276846159. This means that and -7.54276846159 are the real solutions.

If you have solved the problem by rewriting as , you would have lost one of the solutions.

If you would like to test yourself by working some problems similar to this example, click on Problem

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Author: Nancy Marcus