## 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 9.2d:

Answers:        There are an infinite number of solutions: and are the exact solutions, and and are the approximate solutions.

Solution:

To solve for x, first isolate the cosine term.

If we restrict the domain of the cosine function to , we can use the arccos function to solve for x.

The period of is and the period of is The cos x is positive in the firsts and fourth quadrant. This means that the a second solution is

Since the period is this means that the values will repeat every radians. Therefore, the solutions are , and where n is an integer.

These solutions may or may not be the answers to the original problem. You much check them, either numerically or graphically, with the original equation.

Numerical Check:

Check the answer x=2.05824124

• Left Side:

• Right Side:

Since the left side equals the right side when you substitute 2.05824124for x, then 2.05824124 is a solution.

Check the answer x=6.8429379

• Left Side:

• Right Side:

Since the left side equals the right side when you substitute 6.8429379for x, then 6.8429379 is a solution.

Graphical Check:

Graph the equation

Note that the graph crosses the x-axis many times indicating many solutions.

Note the graph crosses at 2.05824124 (one of the solutions). Since the period of the function is , the graph crosses again at 2.05824124+8.901179=10.95942 and again at , etc.

Note the graph crosses at 6.8429379 (one of the solutions). Since the period of the function is , the graph crosses again at 6.8429379+8.901179=15.7441169 and again at , etc.

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Author: Nancy Marcus