##
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.

**
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contents.
**

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