Đặt \(a=\log_35;b=\log_45\). Mệnh đề nào dưới đây luôn đúng?
\(\log_{15}10=\dfrac{a+2ab}{2ab}\).\(\log_{15}10=\dfrac{a^2-ab}{ab}\).\(\log_{15}10=\dfrac{a+2ab}{2\left(ab+b\right)}\).\(\log_{15}10=\dfrac{a^2-ab}{ab+b}\).Hướng dẫn giải:Vì \(a=\log_35;b=\log_45\) nên ta có:
\(b=\log_{2^2}5=\dfrac{1}{2}\log_25\)
=> \(\log_25=2b\)
Và \(\log_23=\log_25.\log_53=2b.\dfrac{1}{a}=\dfrac{2b}{a}\)
Ta tính:
\(\log_{15}10=\log_{15}\left(2.5\right)=\log_{15}2+\log_{15}5\)
\(=\dfrac{1}{\log_215}+\dfrac{1}{\log_515}\)
\(=\dfrac{1}{\log_23+\log_25}+\dfrac{1}{\log_53+\log_55}\)
\(=\dfrac{1}{\dfrac{2b}{a}+2b}+\dfrac{1}{\dfrac{1}{a}+1}\)
\(=\dfrac{a}{2b\left(a+1\right)}+\dfrac{a}{a+1}\)
\(=\dfrac{a+2ab}{2\left(ab+a\right)}\)