Answer
$\log\sqrt{\dfrac{x^{2}+4}{(x^{2}+1)(x^{3}-7)^{2}}}=\dfrac{1}{2}[\log(x^{2}+4)-\log(x^{2}+1)-2\log(x^{3}-7)]$
Work Step by Step
$\log\sqrt{\dfrac{x^{2}+4}{(x^{2}+1)(x^{3}-7)^{2}}}$
Before we begin with the expansion process, let's rewrite the expression like this:
$\log[\dfrac{x^{2}+4}{(x^{2}+1)(x^{3}-7)^{2}}]^{1/2}$
The exponent in this expression can be taken to the front of the logarithm to multiply:
$\log[\dfrac{x^{2}+4}{(x^{2}+1)(x^{3}-7)^{2}}]^{1/2}=\dfrac{1}{2}\log\dfrac{x^{2}+4}{(x^{2}+1)(x^{3}-7)^{2}}=...$
The logarithm of a division can be expanded as a substraction:
$...=\dfrac{1}{2}[\log(x^{2}+4)-\log(x^{2}+1)(x^{3}-7)^{2}]=...$
The logarithm of a product can be expanded as a sum:
$...=\dfrac{1}{2}[\log(x^{2}+4)-[\log(x^{2}+1)+\log(x^{3}-7)^{2}]]=...$
$...=\dfrac{1}{2}[\log(x^{2}+4)-\log(x^{2}+1)-\log(x^{3}-7)^{2}]=...$
The exponent present in $\log(x^{3}-7)^{2}$ can be taken to the front of its respective logarithm to multiply:
$...=\dfrac{1}{2}[\log(x^{2}+4)-\log(x^{2}+1)-2\log(x^{3}-7)]$