${\log ^5} + {\log ^{200}} + {\log ^{500}} + {\log ^{2000}}$$ = {\log ^{\left( {5 \times 200 \times 500 \times 2000} \right)}} = \log _{10}^{\left( {{{10}^9}} \right)} = 9\log _{10}^{10} = 9$