Question

логарифм на фото математика

Answer 1

${6}^{ \frac{2 - lg18}{lg36} } \cdot {14}^{ \frac{1 + lg5}{lg196} } ={(10^{lg6})}^{ \frac{2 - lg18}{lg36} } \cdot {(10^{lg14})}^{ \frac{1 + lg5}{lg196} } = \\ = 10^{ \frac{lg6(2 - lg18)}{lg36} } \cdot {10}^{ \frac{lg14(1 + lg5)}{lg196} } = {10}^{ \frac{2 - lg18}{ \frac{lg36}{lg6} } + \frac{1 + lg5}{ \frac{lg196}{lg14} } } = \\ = {10}^{ \frac{2 - lg18}{\log_6 36} + \frac{1 + lg5}{\log_{14}196} } = {10}^{ \frac{2 - lg18}{\log_6 6^2} + \frac{1 + lg5}{\log_{14}14^2} } = {10}^{ \frac{2 - lg18}{2} + \frac{1+lg5}{2} } = \\ = {10}^{ \frac{2 - lg18 + 1 + lg5}{2} } = ({10}^{3 + lg( \frac{5}{18}) })^{ \frac{1}{2} } = {( {10}^{3} \cdot {10}^{lg( \frac{5}{18} )} )}^{ \frac{1}{2} } =$
$= {(1000 \cdot \frac{5}{18} )}^{ \frac{1}{2} } = {( \frac{2500}{9}) }^{ \frac{1}{2} } = {( \frac{ {50}^{2} }{ {3}^{2} }) }^{ \frac{1}{2} } = ( {( \frac{50}{3} )}^{2})^{ \frac{1}{2} } = \\ = \frac{50}{3} = 16 \frac{2}{3}$