Giải bài 1 trang 25 sách bài tập toán 11 – Chân trời sáng tạo tập 2

Đề bài

Tính giá trị của các biểu thức

a) ${\left( {\frac{{27}}{8}} \right)^{\frac{5}{6}}}.{\left( {\frac{{{4^{\frac{3}{2}}}}}{{{3^3}}}} \right)^{\frac{1}{2}}}$;

b) $\log \sqrt 5  + \log \sqrt 2$;

c) ${\left( {\frac{{16}}{{81}}} \right)^{ – \frac{3}{4}}} + {\log _5}\frac{9}{4} + {\log _5}\frac{4}{9}$;

d) ${\log _2}7.{\log _3}16.{\log _9}3.{\log _7}9$.

Lời giải chi tiết

a) ${\left( {\frac{{27}}{8}} \right)^{\frac{5}{6}}}.{\left( {\frac{{{4^{\frac{3}{2}}}}}{{{3^3}}}} \right)^{\frac{1}{2}}}$ $= {\left( {\frac{3}{2}} \right)^{3.\frac{5}{6}}}{\left( {\frac{{{2^{2.\frac{3}{2}}}}}{{{3^3}}}} \right)^{\frac{1}{2}}}$ $= {\left( {\frac{3}{2}} \right)^{\frac{5}{2}}}.{\left( {\frac{2}{3}} \right)^{\frac{3}{2}}}$ $= {\left( {\frac{3}{2}} \right)^{\frac{5}{2}}}.{\left( {\frac{3}{2}} \right)^{\frac{{ – 3}}{2}}}$ $= {\left( {\frac{3}{2}} \right)^{\frac{5}{2} + \frac{{ – 3}}{2}}}$ $= \frac{3}{2}$;

b) $\log \sqrt 5  + \log \sqrt 2$ $= \log \left( {\sqrt 5 .\sqrt 2 } \right)$ $= \log \sqrt {10}$ $= \log {10^{\frac{1}{2}}}$ $= \frac{1}{2}$;

c) ${\left( {\frac{{16}}{{81}}} \right)^{ – \frac{3}{4}}} + {\log _5}\frac{9}{4} + {\log _5}\frac{4}{9}$ $= {\left( {\frac{2}{3}} \right)^{4.\frac{{ – 3}}{4}}} + {\log _5}\left( {\frac{9}{4}.\frac{4}{9}} \right)$ $= {\left( {\frac{2}{3}} \right)^{ – 3}} + {\log _5}1$ $= {\left( {\frac{3}{2}} \right)^3}$ $= \frac{{27}}{8}$;

d) ${\log _2}7.{\log _3}16.{\log _9}3.{\log _7}9$ $= \frac{{{{\log }_9}7}}{{{{\log }_9}2}}.{\log _7}9.2{\log _3}4.\frac{1}{2}{\log _3}3$ $= \frac{1}{{{{\log }_7}9.{{\log }_9}2}}.{\log _7}9.{\log _3}4$

$= \frac{{2{{\log }_3}2}}{{\frac{1}{2}{{\log }_3}2}}$ $= 4$