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Section 1.8 : Logarithm Functions

9. Write \(\displaystyle {\log _4}\left( {\frac{{x - 4}}{{{y^2}\,\sqrt[5]{z}}}} \right)\) 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. So, here is the work for this problem.

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

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 an 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.