RULES OF LOGARITHMS - Example

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: tex2html_wrap_inline48

Example 5: Expand tex2html_wrap_inline50 and state the domain.

Solution: The domain of tex2html_wrap_inline50 is the set of all real numbers such that

displaymath54

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 tex2html_wrap_inline50 is the set of all real numbers that are in the interval tex2html_wrap_inline78 or tex2html_wrap_inline80 , that is

displaymath82


The domain of tex2html_wrap_inline84 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, tex2html_wrap_inline50 is equivalent to tex2html_wrap_inline84 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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Author: Nancy Marcus

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