Question

If $\frac{\log x}{\log y}=\frac{3}{2}$ and $\log \left(xy\right)=5$, then number of zeros in $x\times y$ are

Answer 1

$\log xy=5...\left(1\right)\log x+\log y=5\text{...(Multiplication rule)}\frac{\log x}{\log y}=\frac{3}{2}...\left(2\right)\therefore \log x=\frac{3}{2}\log ySo,\frac{3}{2}\log y+\log y=5\text{Assuming base 10}\frac{5}{2}{\log }_{10}y=5{\log }_{10}10\log y^{\frac{5}{2}}=\log {10}^{5}So,y={10}^2\log x=\frac{3}{2}\log y\log x=\frac{3}{2}\log {10}^2=\log {10}^{3}So,x=1000y=100$