[tex] ^{5} log \sqrt{2} \times \: \: ^{2} log625 + {}^{3} log27 = [/tex]

[tex] {}^{5} log \sqrt{2} \times {}^{2} log625 + {}^{3} log27 \\ = {}^{5} log {2}^{ \frac{1}{2} } \times {}^{2} log {5}^{4} + {}^{3} log {3}^{3} \\ = \frac{1}{2} {}^{5} log2 \times {}^{2} log {5}^{4} + 3 {}^{3} log3 \\ = \frac{1}{2} {}^{5} log {5}^{4} + 3(1) \\ = ( \frac{1}{2} )(4)(1) + 3 \\ = 5[/tex]