\( \Large log_{ax}x = \frac{1}{log_{x}ax} => log_{ax}x = \frac{1}{1+log_{x}a}\)
If a,b,c are in GP => \( \Large log_{x}a, log_{x}b, log_{x}c\ are\ in\ AP \) => \( \Large 1+log_{x}a,1+log_{x}b,1+log_{x}c\ are\ in\ AP \) => \( \Large \frac{1}{1+log_{x}a}, \frac{1}{1+log_{x}b}, \frac{1}{1+log_{x}c}\ are\ in\ HP \) => \( \Large log_{ax}x,\ log_{bx}x\ log_{cx}x\ are\ in\ HP. \)
