## 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 9.1c:

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

Solution:         To solve for x, first isolate the tangent term.

If we restrict the domain of the tangent function to , we can use the arctangent function to solve for x.

The period of tangent function is This means that the values will repeat every radians. Therefore, the solutions are 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:        20

Since the left side equals the right side when you substitute 1.5458015for x, then 1.5458015 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 1.5458015 (one of the solutions). Since the period of the function is , the graph crosses again at 1.5458015+3.1415924.687394 and again at , etc.

If you would like to review the solution to problem 9.1d, click on solution.

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Author: Nancy Marcus