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### Section 6.2 : Logarithm Functions

14. Write $$\ln \left( {x\sqrt {{y^2} + {z^2}} } \right)$$ 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. Here is that work.

$\ln \left( {x\sqrt {{y^2} + {z^2}} } \right) = \ln \left( x \right) + \ln \left( {{{\left( {{y^2} + {z^2}} \right)}^{\frac{1}{2}}}} \right)$

Note that we converted to root to a fractional exponent at the same time to help with the next step.

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Finally, we need to use Property 7 on the last logarithm to bring the root exponent out of the logarithm. Here is that work.

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

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. Also, in the second logarithm while each term is squared the whole argument is not squared, i.e. it’s not $${\left( {x + y} \right)^2}$$ and so we can’t bring those 2’s out of the logarithm.