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_inline95

Example 6: Expand tex2html_wrap_inline97 and state the domain.

Solution: The domain of tex2html_wrap_inline97 is the set of all real numbers such that

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You can find the domain algebraically as we did in the previous examples or you can graph

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and notice for what values of x is the graph located above the x-axis. The graph rises above the x-axis to the left of - 1 and to the right of + 1. Therefore the domain of the initial expression is the set of real numbers x such that x < - 1 or x > + 1.
The expression tex2html_wrap_inline97 can be simplified to

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The expression

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and this can be written as

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as long as each term is valid.
For tex2html_wrap_inline119 to be valid, we must have x > 1. For tex2html_wrap_inline123 to be valid, we must have x> - 1. Both terms tex2html_wrap_inline127 and tex2html_wrap_inline129 are always valid because the expressions tex2html_wrap_inline131 and tex2html_wrap_inline133 are always positive. Therefore, if we restrict the domain to all real numbers greater than 1, the expansion of tex2html_wrap_inline135 is valid.

The expression tex2html_wrap_inline137 can be simplified to

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The expression tex2html_wrap_inline141 can be written as tex2html_wrap_inline143 as long as each term is valid.
For tex2html_wrap_inline145 to be valid, we must have x > 2. For tex2html_wrap_inline149 to be valid, we must have x > 3. Both terms are valid when x > 3. Therefore, if we restrict the domain to all real numbers greater than 3, the expansion of is valid.

And we can say that

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as long as we restrict the domain to the set of real numbers greater than 3.

Check: Pick any number in the domain, say x = 10. Find the value of the original expression when x = 10. The value of the original expression tex2html_wrap_inline97 when x = 10 is

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The value of the final expression

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when x = 10 is

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Since both values are the same when x = 10, the original expression and the final expression are equivalent at x = 10. We can also say that for all values of x greater than 3, the original expression is equivalent to the final expression.

If you would like to work a problem, click on Problem.

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Author: Nancy Marcus

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