First of all observe that
[tex]r^6s^3=(r^2s)^3 \implies \sqrt[3]{r^6s^3}=\sqrt[3]{(r^2s)^3}=r^2s[/tex]
So, we have
[tex]\log(r^5s^6\sqrt[3]{r^6s^3})=\log(r^5s^6\cdot r^2s)[/tex]
We can simplify the input as
[tex]\log(r^5s^6\cdot r^2s)=\log(r^7s^7)[/tex]
The logarithm of a multiplication is the sum of the logarithms:
[tex]\log(r^7s^7)=\log(r^7)+\log(s^7)[/tex]
Finally, invoke the logarithm property [tex]\log(a^b)=b\log(a)[/tex]
to get the final answer
[tex]\log(r^7)+\log(s^7)=7\log(r)+7\log(s)[/tex]
