How do you rewrite this Logarithmic problem in expanded form?

enter image source here

1 Answer
May 11, 2018

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

Explanation:

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