**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:**

**Example 4: ** Expand and write the answer in terms of base *e*.

**Solution:** Write as using Rule 2.

Now expand further by using Rule 1.

Now convert the above terms to the base *e*.

as long as *x* > 0, *y* > 0, *z* > 0, and *w* > 0.

Since

the two expressions are equivalent. This means that if you substitute the same values of x, y, w, and z in both expressions, the results will be equal.

Let's check the answer by letting x = 2, y = 3, w = 4, and z =5. Substituting these values in the original problem yields

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

Since the check works, you have successfully worked this problem.

**If you would like to review another example, click on Example.**

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