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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