[tex]log\frac{5\cdot10^{-3}}{2\cdot10}}=log\left(\frac{5}{2}\cdot\frac{10^{-3}}{10}\right)=log(2.5\cdot10^{-3-1}})=log(2.5\cdot10^{-4})=(*)\\\\use\ log(a\cdot b)=loga+logb\ and\ loga^b=b\cdot loga\\\\(*)=log2.5+log10^{-4}=log2.5-4log10=log2.5-4\\\vdots\\log2.5=log\frac{25}{10}=log25-log10=log25-1=log5^2-1=2log5-1\\\\therefore\\\\(*)=2log5-1-4=2log5-5[/tex]