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_inline132

Example 4: Expand tex2html_wrap_inline134 and write the answer in terms of base e.

Solution: Write tex2html_wrap_inline134 as tex2html_wrap_inline140 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.


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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Author: Nancy Marcus

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