## 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 2: The above equation is valid only if all of the terms are valid. The first term is valid if 3x- 5 > 0 or the second term is valid if 2x- 3 > 0 or and the third term is valid if x > 0. Therefore, the equation is valid when all three 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 if then a=b . Therefore, if  Solve for x.      The exact answer is and the approximate answer is Check the answer by substituting 8.63104367407 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 8.63104367407 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 8.63104367407. This means that 8.63104367407 is the real solution.

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