Question

[(1-log(6)3)^2+log(6)2 *log(6)18]/log(6)4

Answer 1

换底公式:log以a为底b的对数=lgb/lga(lga是log以10为底a的对数).公式lg(a*b)=lga+lgb,lg(a/b)=lga-lgb,lg(a^b)=b*lga.

原式=[(1-lg3/lg6)^2+(lg2/lg6)*(lg18/lg6)]/(lg4/lg6)

=[1-(2*lg3)/lg6+(lg3)^2/(lg6)^2+(lg2*lg18)/(lg6)^2]*(lg6/lg4)

=[1-(2*lg3)/lg6+(lg3^2+lg2*lg18)/(lg6)^2]*(lg6/lg4)

=[1-(2*lg3)/lg6+(lg3^2+lg2*(lg2+2*lg3)))/(lg6)^2]*(lg6/lg4)

=[1-(2*lg3)/lg6+(lg3+lg2)^2/(lg6)^2]*(lg6/lg4)

=[1-(2*lg3)/lg6+(lg6)^2/(lg6)^2]*(lg6/lg4)

=[1-(2*lg3)/lg6+1]*(lg6/lg4)

=2*(lg6/lg4)-(2*lg3)/lg4

=(2*lg6)/(2*lg2)-(2*lg3)/(2*lg2)

=lg6/lg2-lg3/lg2

=(lg6-lg3)/lg2=lg(6/3)/lg2=lg2/lg2=1