## 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 8.2c:

Solution:

Note that the domain of is the set of real numbers such that 5 - 2x > 0, or when because you cannot take the log of zero or a negative number. If any of your answers are greater than or equal to , you must discard them as extraneous solutions.

Let's start the process to find the solution to the equation.

Isolate the logarithmic term.

Convert the logarithmic equation to an exponential equation with base e.

Solve for x by isolating x.

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 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 . This means that is the real solution.

If you would like to review the solution to problem 8.2d, click on Solution

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