If a, b, c are in geometric progression, then log_ax x, log_bx x and log_cx x are in

Concept:

If a, b, c are in geometric progression then b² = ac

If b - a = c - b, than a, b, c are in AP.

If 1/a, 1/b, 1/c are in AP than a, b, c are in HP.

\({\log _a}b = \frac{1}{{{{\log }_b}a}}\)

Calculation:

If a, b, c are in geometric progression then b2 = ac

So, by multiplying both side by x² and taking log on both side to the base x

\({\log _x}({x^2}{b^2}) = {\log _x}({x^2}ac)\)

\({\log _x}({x^2}{b^2}) = {\log _x}(xa\cdot xc)\)

\(2{\log _x}xb = {\log _x}xa + {\log _x}xc\)

\({\log _x}xb - {\log _x}xa = {\log _x}xc - {\log _x}xb\)

so logx ax, logx bx and logx cx are in AP.

\(\frac{1}{{{{\log }_{ax}}x}},\frac{1}{{{{\log }_{bx}}x}},\frac{1}{{{{\log }_{cx}}x}}\) are also in AP

\({\log _{ax}}x,{\log _{bx}}x,{\log _{cx}}x\) are in HP