## 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:

The above equation is valid only if all of the terms are valid. The first term is valid if the second term is valid if the third term is valid if , and the fourth term is valid if Therefore, the equation is valid when all four of these conditions are met, or when The domain is the set of real numbers greater than .

Simplify both sides of the equation using the rules of logarithms.

Recall that when a = b. Therefore, if

Solve for x.

The exact answers are and the approximate answers are 1.69771560359 and -37.6977156036. Neither of these answers is located in the restricted domain. Therefore, there are no real solutions to this problem.

If you did not make this observation, you can catch it in the check.

Check the answer by substituting 1.69771560359 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 the empty set, then 1.69771560359 is not a solution.

Check the answer by substituting -37.6977156036 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 the empty set, then -37.6977156036 is not 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 never crosses the x-axis. This means that there are no real 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