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Section 6.2 : Logarithm Functions

13. Write \(\log \left( {3{x^4}{y^{ - 7}}} \right)\) 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 5 to break up the product into individual logarithms. Note that just because the property only has two terms in it does not mean that it won’t work for three (or more) terms. Here is the application of Property 5.

\[\log \left( {3{x^4}{y^{ - 7}}} \right) = \log \left( 3 \right) + \log \left( {{x^4}} \right) + \log \left( {{y^{ - 7}}} \right)\] Show Step 2

Finally, we need to use Property 7 on the last two logarithms to bring the exponents out of the logarithms. Here is that work.

\[\log \left( {3{x^4}{y^{ - 7}}} \right) = \require{bbox} \bbox[2pt,border:1px solid black]{{\log \left( 3 \right) + 4\log \left( x \right) - 7\log \left( y \right)}}\]

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 product.