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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 8.6d:**

**Answers:**T
he exact answers are
The approximate answers are
and

**Solution:**

The above equation is valid only if
is valid. The term
is valid if
or
Therefore, the equation is valid when the
domain is the set of real numbers less than
or
greater than

Covert the logarithmic equation to an exponential equation with base *e*.

The exact answers are
The approximate answers are
and
-0.979582.

These answers may or may not be the solutions to the original equation. You
must check them in the original equation, either by numerical substitution
or by graphing.

**Numerical Check:**

Check the answer
by substituting 2.729582 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 2.729582 for x, then
*x*=2.729582 is a solution.
Check the answer
by substituting -0.979582 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 -0.979582 for x, then
*x*=-0.979582 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 2.729582 and -0.979582. This means that 2.729582 and -0.979582 are the real solutions.
If you have trouble graphing the above problem, you might try graphing the
equivalent function

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