Question

find the value of
[1] log₁₀5 * log₁₀​20 + (log₁₀2)²
[2] 5^{log_(5^1/2)2} + 9^log₃7 + 8^log₂5

Answer 1

$$\begin{gathered} \mathrm{Dear}\mathrm{Student}, \\ {\log }_{10}5\times {\log }_{10}20+{\left({\log }_{10}2\right)}^2 \\ ={\log }_{10}5\times {\log }_{10}\left(2\times 10\right)+{\left({\log }_{10}2\right)}^2 \\ ={\log }_{10}5\times \left(1+{\log }_{10}2\right)+{\left({\log }_{10}2\right)}^2 \\ ={\log }_{10}5+{\log }_{10}5{\log }_{10}2+{\left({\log }_{10}2\right)}^2 \\ ={\log }_{10}5+{\log }_{10}2\left[{\log }_{10}5+\left({\log }_{10}2\right)\right] \\ ={\log }_{10}5+{\log }_{10}2\left[{\log }_{10}\left(5\times 2\right)\right] \\ ={\log }_{10}5+{\log }_{10}2\left[{\log }_{10}10\right] \\ ={\log }_{10}5+{\log }_{10}2 \\ ={\log }_{10}\left(5\times 2\right) \\ =\left[{\log }_{10}10\right] \\ =1 \\ 2) \\ {5}^{{\log }_{5^{\frac{1}{2}}}2}+{\left(3^2\right)}^{{\log }_{3}7}+{\left(2^3\right)}^{{\log }_{2}5} \\ =5^{2{\log }_{5}2}+{\left(3\right)}^{2{\log }_{3}7}+{\left(2\right)}^{3{\log }_{2}5} \\ =5^{{\log }_5{2}^2}+{\left(3\right)}^{{\log }_3{7}^2}+{\left(2\right)}^{{\log }_2{5}^3} \\ =2^2+7^2+5^3 \\ =4+49+125 \\ \mathrm{Regards} \end{gathered}$$