## 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 functions.

Solve for x in the following equation.

Problem 8.5a:

Solution:

The above equation is valid only if each of the three terms is valid. The term is valid if x>2. The term is valid if x>-4. The term is valid if x>1. Therefore, the equation is valid when the domain is the set of real numbers is greater than2, greater than -4, and greater than1. This means that the equation is valid if we restrict the domain to the set of real numbers greater than2.

Simplify the equation and solve.

These answer may or may not be the solution to the original equation. You must it in the original equation, either by numerical substitution or by graphing.

Numerical Check:

• 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:

The left side does not equal the right side exactly because we rounded the numbers. However, it is close enough to verify our answers.Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 2.192582 for x, then x=2.192582is a solution.

Graphical Check:

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. You may have to modify the equation for your calculator first. Rewrite f(x) as

Note that the graph crosses the x-axis at 2.192582. This means that 2.192582 is the real solution.

If you would like to review the solution to problem 8.5b, click on solution.

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Author: Nancy Marcus