等差数列证明题已知logaX,logbX,logcX成等差数列,且X不等于1,试证:1/lga,1/lgb,1/lgc成等差数列

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等差数列证明题已知logaX,logbX,logcX成等差数列,且X不等于1,试证:1/lga,1/lgb,1/lgc成等差数列
等差数列证明题
已知logaX,logbX,logcX成等差数列,且X不等于1,试证:1/lga,1/lgb,1/lgc成等差数列

等差数列证明题已知logaX,logbX,logcX成等差数列,且X不等于1,试证:1/lga,1/lgb,1/lgc成等差数列
logaX,logbX,logcX成等差数列
logcX-logbX=logbX-logaX
2logbX=logaX+logcX
两边同取以10为底的对数:
2lgx/lgb=lgx(1/lga+1/lgc)
2/lgb=1/lga+1/lgc
1/lgc-1/lgb=1/lgb-1/lga
1/lga,1/lgb,1/lgc成等差数列

用换底公式,1/lga=loga10/(loga)a=loga10
1/lgb=logb10/(logb)b=logb10
1/lgc=logc10/(logc)c=logc10
若x=10,则得证