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 9.1d:

1.875 sin (x)-0.684=0

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

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

If we restrict the domain of the sine function to , we can use the arcsine function to solve for x.

The sine function is positive in the first quadrant and the second quadrant. The angle is a reference angle that terminates in the first quadrant. The angle that terminates in the second quadrant that has a reference angle is

The period of sine function is This means that the values will repeat every radians. Therefore, the exact solutions are and where n is an integer. The approximate solutions are and

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=0.37341799

• Left Side:

• Right Side:        0

Since the left side equals the right side when you substitute 0.37341799for x, then 0.37341799 is a solution.

Check the answer x=2.768175

• Left Side:

• Right Side:        0

Since the left side equals the right side when you substitute 2.768175 for x, then 2.768175 is a solution.

Graphical Check:

Graph the equation

f(x)=1.875sin(x)-0.684.

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

Note the graph crosses at 0.37341799 (one of the solutions) and at 2.768175. Since the period of the function is , the graph crosses again at and again at , etc.

If you would like to go back to the equation table of contents, click on contents.

This site was built to accommodate the needs of students. The topics and problems are what students ask for. We ask students to help in the editing so that future viewers will access a cleaner site. If you feel that some of the material in this section is ambiguous or needs more clarification, or you find a mistake, please let us know by e-mail.

[Algebra] [Trigonometry]
[Geometry] [Differential Equations]
[Calculus] [Complex Variables] [Matrix Algebra]

Author: Nancy Marcus

Copyright © 1999-2019 MathMedics, LLC. All rights reserved.
Contact us
Math Medics, LLC. - P.O. Box 12395 - El Paso TX 79913 - USA
users online during the last hour