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Section 6.2 : Logarithm Functions

16. Combine \(2{\log _4}x + 5{\log _4}y - \frac{1}{2}{\log _4}z\) into a single logarithm with a coefficient of one.

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Hint : The properties that we use to break up logarithms can be used in reverse as well.
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To convert this into a single logarithm we’ll be using the properties that we used to break up logarithms in reverse. The first step in this process is to use the property,

\[{\log _b}\left( {{x^r}} \right) = r{\log _b}x\]

to make sure that all the logarithms have coefficients of one. This needs to be done first because all the properties that allow us to combine sums/differences of logarithms require coefficients of one on individual logarithms. So, using this property gives,

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

Now, there are several ways to proceed from this point. We can use either of the two properties.

\[{\log _b}\left( {xy} \right) = {\log _b}x + {\log _b}y\hspace{0.25in}\hspace{0.25in}{\log _b}\left( {\frac{x}{y}} \right) = {\log _b}x - {\log _b}y\]

and in fact we’ll need to use both in the end. We can use the product property on the first two logarithms (because they are a sum of logarithms) or the quotient property on the last two logarithms (because they are a difference of logarithms).

Which we use first does not matter as we’ll end up with the same result in the end. For this problems we’ll first use the product property on the first two logarithms to get,

\[2{\log _4}x + 5{\log _4}y - \frac{1}{2}{\log _4}z = {\log _4}\left( {{x^2}{y^5}} \right) - {\log _4}\left( {\sqrt z } \right)\] Show Step 3

Finally, we can see that we have a difference of two logarithms left and so we’ll use the quotient property to combine these to get,

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

Note that the only reason we converted the fractional exponent to a root was to make the final answer a little nicer.