# RULES OF LOGARITHMS - Problem 7

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 3: .

Problem 7: Simplify the following term completely

State the domain that makes your final answer equal to the original expression.

There are many ways to work the above problem; we have included one of the ways.

Solution:

Step 1: The expression is of the form which can be simplified to the form or

Step 2: The expression can be simplified to the form or

which can be written as

Step 3: This last expression is of the form which can be simplified to

or in term of our problem

can be written as

Step 4: Both of these last terms are in the form which can be simplified to the form

or in terms of our problem

can be written as

Step 5: These terms can be simplified to

and again to

Step 6: The first term will work if x > 0, the second term will work if x > -5, the third term will work if x > 8, the fourth term will always work, the fifth term will work if x > -9, and the last term will work if x > 4. If we choose the domain as x > 8, then the original expression

is equal to the final expression

Step 7: You can check answer by graphing both expression over the domain x >8. If you see just one graph, you have worked the problem correctly.
Step 8: You can also check the answer by assigning x a value greater than 8 and substituting that value in both expressions. If the answers come out the same, you are correct. Let's let x = 10 in the original equation. If x = 10, the original equation has a value equal to

Let's let x = 10 in the final equation. If x = 10, the final equation has a value equal to

Step 9: Pat yourself on the back because you have correctly worked the problem.

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