##
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 function.

**Solve for x in the following equation.**

**Example 1:**

ln
.
The above equation is valid only if

The domain is the set of real numbers less than -2 or greater than 8.
Convert the equation to an exponential equation with base *e*.

The exact answers are
and the approximate answers
are 16.16864 and
-10.16864.
These answers may or may not be the solutions. You must check them with the
original equation, either by a numerical substitution or by graphing.

**Numerical Check:**

Check the answer
by substituting 16.16864 in
the original equation for x. If the left side of the equation equals the
right side of the equation after the substitution, you have found the
correct answer.

Since the left side of the original equation is equal to the right side of
the original equation after we substitute the value 16.16864 for x, then
*x*=16.16864 is a solution.

Check the answer
by substituting -10.16864 in
the original equation for x. If the left side of the equation equals the
right side of the equation after the substitution, you have found the
correct answer.

Since the left side of the original equation is equal to the right side of
the original equation after we substitute the value -10.16864 for x, then
*x*=-10.16864 is a solution.

**Graphical Check:**

You can also check your answer by graphing

(formed by subtracting the right side of the
original equation from the left side). Look to see where the graph crosses
the x-axis; that will be the real solution. Note that the graph crosses the
x-axis at 16.16864 and -10.16864. This means that 16.16864 and -10.16864 are the real solutions.

**
If you would like to work another example, click on example.
**

If you would like to test yourself by working some problems similar to this
example, click on problem.

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

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

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

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