## 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 9.2b:

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 sine term.

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

The sine of x is positive in the first quadrant and the second quadrant. This means that there are two solutions in the first counterclockwise rotation from 0 to . One angle x terminates in the first quadrant and the second angle terminates in the second quadrant. One solution is

The period of is , and the period of is As 6x rotates radians, x rotates Therefore,the 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:

• Left Side:

• Right Side:

Since the left side equals the right side when you substitute 0.08165955for x, then 0.08165955 is a solution.

• Left Side:

• Right Side:

Since the left side equals the right side when you substitute 0.44193922for x, then 0.44193922 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 0.08165955 ( one of the solutions ). Since the period of the function is , the graph crosses again at 0.08165955+1.04719755=1.128857 and again at , etc.

The graph also crosses at 0.44193922 ( another solution we found ). Since the period is , it will crosses again at 0.44193922+1.04719755=1.48913677 and at , etc

If you would like to review the solution to problem 9.2c, click on solution.

If you would like to go back to the previous section, click on previous

If you would like to go to the next section, click on next

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Author: Nancy Marcus