Question

设log以ab为底的a的对数=p,用p表示log以ab为底的根号(a/b)的对数=

Answer 1

对已知条件log(ab)(a)=p,用10换底得
[lg(a)]/[lg(ab)]=p
[lg(a)]/[lg(a)+lg(b)]=p
解得lg(b)=[(1-p)/p]lg(a)
对所求的式子也用10换底得
log(ab)[√(a/b)]
=[lg√(a/b)]/[lg(ab)]
=(1/2)[lg(a/b)]/[lg(ab)]
=(1/2)[lga-lgb]/[lga+lgb]
=(1/2){lga-[(1-p)/p]lga}/{lga+[(1-p)/p]lga}
=(1/2)[1-(1-p)/p]/[1+(1-p)/p]
=p-(1/2)