已知a=㏒54 27是真数,54的b次方=2,用a,b表示㏒108 81为真数的值

问题描述:

已知a=㏒54 27是真数,54的b次方=2,用a,b表示㏒108 81为真数的值

换底公式log(a)(N)=log(b)(N)/log(b)(a)= lnN/lna=lgN/lga
都换成以ln为底。27=3×3×3,54=2×27,81=3×27,108=2×54,
log(a)(MN)=log(a)(M)+log(a)(N);
答案为4a/3a+6b

a=㏒54 27,54的b次方=2,则b=log54 2;
a=lg27/lg54=3lg3/lg54;b=lg2/lg54;
所以:lg3=(alg54)/3;lg2=blg54;
log108 81=lg81/lg108
=4lg3/(lg54+lg2)
=(4/3)(alg54)/(lg54+blg54)
=(4/3)(alg54)/(1+b)lg54
=4a/3(b+1)
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