Câu 42:
\(log_{abc}\left(ab^2c^3\right)\)
\(=log_{abc}\left(abc\cdot bc^2\right)\)
\(=1+log_{abc}\left(bc^2\right)\)
\(=1+log_{abc}b+log_{abc}c^2\)
\(=1+\dfrac{1}{log_babc}+2\cdot log_{abc}c\)
\(=1+\dfrac{1}{log_ba+1+log_bc}+\dfrac{2}{log_cabc}\)
\(=1+\dfrac{1}{1+log_ba+log_bc}+\dfrac{2}{log_ca+log_cb+1}\)
\(=1+\dfrac{1}{1+\dfrac{1}{m}+\dfrac{log_ac}{log_ab}}+\dfrac{2}{\dfrac{1}{log_ac}+\dfrac{log_ab}{log_ac}+1}\)
\(=1+\dfrac{1}{\dfrac{m+1}{m}+\dfrac{n}{m}}+\dfrac{2}{\dfrac{1}{n}+\dfrac{m}{n}+1}\)
\(=1+\dfrac{m}{m+1+n}+\dfrac{2}{\dfrac{1+m+n}{n}}\)
\(=1+\dfrac{m}{m+n+1}+\dfrac{2n}{m+n+1}=\dfrac{m+2n+m+n+1}{m+n+1}\)
\(=\dfrac{2m+3n+1}{m+n+1}\)
=>Chọn A
