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Section 1.8 : Logarithm Functions

8. Write \(\ln \left( {x\sqrt {{y^2} + {z^2}} } \right)\) in terms of simpler logarithms.

Show Solution

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.

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

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 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.