## 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.

Problem 1:

Solution:

The above equation is valid only if all of the terms are valid. The first term is valid if x>0, the second term is valid if x > -2, and the third term is valid if Therefore, the equation is valid when all three of these conditions are met, or when x > 0. The domain is the set of real numbers greater than 0.

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

Recall that if that a = b. Therefore, if

Solve for x.

Check the answer x = 2 by substituting 2 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 2 for x, then x = 2 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 2. This means that 2 is the real solution.

If you would like to review the solution to problem 8.3b, click on solution.

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Author: Nancy Marcus