How do you rewrite this Logarithmic problem in expanded form?

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Answer 1 · 2018-05-11T21:34:18

#=7/2logx-5/2logy-7/2logz#

#logsqrt(x^7/(y^5z^7))=log(x^7/(y^5z^7))^(1/2)#

Recall that #log(x^a)=alogx#, so

#log(x^7/(y^5z^7))^(1/2)=1/2log(x^7/(y^5z^7))#

Furthermore, recalling that #log(a/b)=loga-logb,#

#1/2log(x^7/(y^5z^7))=1/2[log(x^7)-log(y^5z^7)]#

Recalling that #log(ab)=loga+logb,#

#1/2[log(x^7)-log(y^5z^7)]=1/2[log(x^7)-(log(y^5)+log(z^7))]#

Apply the exponent property to all remaining logarithms and distribute the negative through:

#=1/2[7logx-5log(y)-7log(z)]#

Distribute the #1/2:#

#=7/2logx-5/2logy-7/2logz#