**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:**

**Problem 5: ** Simplify 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
must be greater than zero.

The expression can be simplified to

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

The expression can be written as

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.

can be written in terms of base 10 as

**Check:** Since is equivalent to

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

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

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:

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