**Let a be a positive number such that a does not equal 1, let
n be a real number, and let u and v be positive real numbers.**

**Logarithmic Rule 2:**

**Example 5: ** Expand and state the domain.

**Solution:** The domain of is the set of all real numbers such that

Either *x*-7>0 and *x*+4>0 or *x*-7<0 and *x*+4<0 .

This means that if *x* > 7 and *x* > - 4, both the numerator and the
denominators are positive and the value of the fraction is positive. If we
restrict the values of x to be greater than 7, they are automatically
greater than - 4 and the domain can be simplified to *x* > 7.

**or**

This means that if *x* > 7 and *x* < -4, both the numerator and the
denominator are negative, and the value of the fraction is positive. If we
restrict the values of x to be less than - 4, they are automatically less
than 7, and the domain can be simplified to *x* < -4.

The domain of is the set of all real numbers that are in the interval **or** , that is

The domain of is the set of
real numbers that works for both. The real number *x* > 7 works for the
first term and the real number *x* > -4 works for the second term. You need a set of numbers that will work for both terms. The domain is the set of real numbers where *x* > 7.

Therefore, is equivalent to only when the domain is restricted to those real numbers *x* > 7.

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

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