RULES OF LOGARITHMS - Problem 5

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_inline111

Problem 5: Simplify tex2html_wrap_inline113 and write the answer in terms of a base 10. What assumptions must be made about a, b, d, and d before you can work this problem?

Solution: To work the problem as it is, the value of the expression tex2html_wrap_inline115 must be greater than zero.

The expression tex2html_wrap_inline113 can be simplified to

displaymath119

as long as the value of the expressions abc and 2de are both positive.

The expression tex2html_wrap_inline121 can be written as

displaymath123

as long as a, b, c, d, and e are all positive.

If the values of a, b, c, d, and e are all positive, then we can expand the original expression to the last expression. If the values are not all positive, then we cannot expand the original expression.

displaymath123

can be written in terms of base 10 as

displaymath127


Check: Since tex2html_wrap_inline113 is equivalent to

displaymath131

let us choose values for a, b, c, d, and e and substitute them into the original expression and in our final expression, the answers should be equal. Let us try it.

Suppose a = 2, b = 3, c = 4, d = 5, and e = 6, the original expression has a value

displaymath133

Now let's substitute these same values in the final expression.

displaymath135

Both answers are the same, therefore the original expression is equivalent to the final expression as long as a, b, c, d, and e are all positive numbers.

A last check using our numbers:

displaymath137

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Author: Nancy Marcus

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