If x is a positive real nunber different from unity such that 2logx to the base b=log x to the base a +log x to the base c, then prove that c^2=(ca)^(log b to the base a)?

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Answer 1 · 2017-07-12T16:40:11

Given

#2log_bx=log_ax+log_cx#

#=>2/log_xb=1/log_xa+1/log_xc#

#=>2/log_xb=(log_xa+log_xc)/(log_xa xxlog_xc)#

#=>2/log_xb=log_x(ac)/(log_xa xxlog_xc)#

#=>2log_xc=log_x(ac)xx(log_ax xxlog_x b)#

#=>2log_xc=log_x(ac)xxlog_ab#

#=>log_xc^2=log_ab(log_x(ac))#

#=>log_xc^2=log_x(ac)^(log_ab)#

#=>c^2=(ac)^(log_ab)#