Arithematic Progression -Logarithm

If $\displaystyle log_k x, log_m x , log_nx$ are in A.P then prove that $\displaystyle n^2 = (kn)^{log_km}$

If the terms are in A.P then : $\displaystyle 2 log_mx = log_k x + log_nx = \frac{2}{log_xm} = \frac{1}{log_xk}+ \frac{1}{log_xn}$

=$\displaystyle \frac{2}{log_xm}= \frac{log_xn+log_xk}{log_xklog_xn}$ can we solve this way or not..............please guide

Re: Arithematic Progression -Logarithm

$\displaystyle 2\log_{m}x = \log_{k}x + \log_{n}x.$

Start by using the change of base formula so that all logs are to base $\displaystyle k.$

Cancel the $\displaystyle \log_{k}x$ throughout, cross multiply and that gets you a $\displaystyle 2\log_{k}n$ (which becomes) $\displaystyle \log_{k}n^2$ on the LHS.

Try finishing from there.

Re: Arithematic Progression -Logarithm

thanks a lot....i got it..regards,sachin