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Section 6.2 : Logarithm Functions

15. Write \({\log _4}\left( {\displaystyle \frac{{x - 4}}{{{y^2}\,\sqrt[5]{z}}}} \right)\) in terms of simpler logarithms.

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So, we’re being asked here to use as many of the properties as we can to reduce this down into simpler logarithms.

First, we can use Property 6 to break up the quotient into two logarithms. Here is that work.

\[{\log _4}\left( {\frac{{x - 4}}{{{y^2}\,\sqrt[5]{z}}}} \right) = {\log _4}\left( {x - 4} \right) - {\log _4}\left( {{y^2}\,{z^{\frac{1}{5}}}} \right)\] Show Step 2

Next, we need to use Property 5 to break up the product in the second logarithm into two logarithms.

\[{\log _4}\left( {\frac{{x - 4}}{{{y^2}\,\sqrt[5]{z}}}} \right) = {\log _4}\left( {x - 4} \right) - \left( {{{\log }_4}\left( {{y^2}} \right) + {{\log }_4}\left( {{z^{\frac{1}{5}}}} \right)} \right)\]

Be careful with the minus sign that was in front of the second logarithm from Step 1! Because of that we need to have parenthesis on the product once we use Property 5. The sum of the two “smaller” logarithms is the same as the product logarithm from Step 1 and so because we have a minus sign in front of the product logarithm we also need to have a minus sign in front of the two logarithms after using Property 5. The only way to make sure of this is to use the parenthesis as shown.

Show Step 3

Finally, we’ll distribute the minus sign through the parenthesis and then use Property 7 on the last two logarithms to bring the exponents out of the logarithms. Here is that work.

\[{\log _4}\left( {\frac{{x - 4}}{{{y^2}\,\sqrt[5]{z}}}} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{{{\log }_4}\left( {x - 4} \right) - 2{{\log }_4}\left( y \right) - \frac{1}{5}{{\log }_4}\left( z \right)}}\]

Remember that we can only bring an exponent out of a logarithm if is on the whole argument of the logarithm. In other words, we couldn’t bring any of the exponents out of the logarithms until we had dealt with the quotient and product. Recall as well that we can’t split up a sum/difference in a logarithm. Finally, make sure that you are careful in dealing with the minus sign we get from breaking up the quotient when dealing with the product in the denominator.