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 4:**

The above equation is valid only if
or *x*<-4or *x*>9. The domain is the set of real numbers

Simplify the left side of the equation using the rules of logarithms.

Convert the equation to an exponential equation with base 102.

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
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.

- Left Side:

- Right Side:

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

Check the answer
by substituting
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.

- Left Side:

- Right Side:

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value for x, then 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
.
This means that
are the real solutions.

You may have to change the original equation somewhat to graph it because most graphing calculators only have the natural log function and the common log function. Rewrite the original equation in the equivalent form and graph it

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

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