##
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*)=1.875*sin*(*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]
S.O.S. MATHematics home page

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