Chứng minh: log23(a2b)=4log23a−1−log3a−2.logb2+log23b

Question

Chứng minh: \(\log_3^2(a^2b)=4\log_3^2a^{-1}-\log_3a^{-2}.\log{b^2}+\log_3^2{b}\)

Answer 1

Ta có: VP=\(4\log_3^2a^{-1}-\log_3a^{-2}.\log{b^2}+\log_3^2{b}\)

\(=4\left [-\log_3a \right ]^2-(-2)\log_3a.2\log{b}+\log_3^2{b}\)

\(=4\log_3^2a +4\log_3a.\log{b}+\log_3^2{b}\)

\(=\left ( 2\log_3a+\log_3b \right )^2\)

\(=\left ( \log_3a^2+\log_3b \right )^2\)

\(=[ \log_3({a^2b )}]^2\)=VP

Vậy ta có đpcm