Question

Given $\log 2$ and $\log 3$, find $\log \sqrt[3]{3\left(\frac{3^2{5}^4}{\sqrt{2}}\right)}$.

Answer 1

$\log \sqrt[3]{3\left(\frac{3^2{5}^4}{\sqrt{2}}\right)}=\frac{1}{3}(\log 3^2+\log 5^4-\frac{1}{2}\log 2)=\frac{1}{3}(2\log 3+4\log 5-\frac{1}{2}\log 2)=\frac{1}{3}(2\log 3+4(1-\log 2)-\frac{1}{2}\log 2)=\frac{1}{3}(2\log 3-\frac{9}{2}\log 2+4)\log 2=.3010300,\log 3=.4771213\Rightarrow \frac{1}{3}(2\log 3-\frac{9}{2}\log 2+4)=1.1998692$