Ta cần chứng minh
$1<\log_{ab}c+\log_{bc}a+\log_{ac}b<2$
$\Leftrightarrow 1<\frac{1}{\log_c{ab}}+\frac{1}{\log_a{bc}}+\frac{1}{\log_b{ac}}<2$
$\Leftrightarrow 1<\frac{1}{\log_c{a}+\log_c{b}}+\frac{1}{\log_a{b}+\log_a{c}}+\frac{1}{\log_b{a}+\log_b{c}}<2$
$\Leftrightarrow 1<\frac{1}{\log_c{a}+\log_c{b}}+\frac{1}{\log_a{b}+\log_a{c}}+\frac{1}{\log_b{a}+\log_b{c}}<2$
Trong đó $x=\log_a{b},y=\log_b{c},z=\log_c{a}\Rightarrow \begin{cases}x,y,z>0 \\ xyz=1 \end{cases}$.
$\Leftrightarrow 1<\frac{1}{z+\frac{1}{y}}+\frac{1}{x+\frac{1}{z}}+\frac{1}{y+\frac{1}{x}}<2$
$\Leftrightarrow 1<\frac{y}{yz+1}+\frac{z}{xz+1}+\frac{x}{xy+1}<2$
Nhưng đây là điều không thể xảy ra vì với $x=3,y=3,z=\frac{1}{9}$ thì $\frac{y}{yz+1}+\frac{z}{xz+1}+\frac{x}{xy+1}=\frac{79}{30}>2$.