Question

高一计算

计算，﹙lg5﹚²＋lg2×lg50＋﹙以5为底√2的对数•以7为底√9的对数﹚／﹙以5为底1/3的对数•以7为底³√4的对数﹚
请详写过程，谢谢。

Answer 1

﹙lg5﹚²＋lg2×lg50＋﹙以5为底√2的对数•以7为底√9的对数﹚／﹙以5为底1/3的对数•以7为底³√4的对数﹚

先算前面﹙lg5﹚²＋lg2×lg50部分
=lg5*lg5+lg2*(lg5+lg10)
=lg5*lg5+lg2*lg5+lg2
=lg5(lg5+lg2)+lg2
=lg5+lg2
=1
再算后面部分﹙以5为底√2的对数•以7为底√9的对数﹚／﹙以5为底1/3的对数•以7为底³√4的对数﹚
=(log5(√2)*log7(√9))/(log5(1/3)*log7(³√4))
=((lg√2/lg5)*(lg√9/lg7))/((lg(1/3)/lg5)*(lg³√4/lg7))
=(lg√2*lg√9)/(lg(1/3)*lg³√4)
=(1/2lg2*lg3)/(-lg3*(2/3)lg2)
=(1/2)/(-2/3)
=-3/4
所以答案是1-3/4=1/4

Answer 2

﹙lg5﹚²＋lg2×lg50＋﹙以5为底√2的对数•以7为底√9的对数﹚／﹙以5为底1/3的对数•以7为底³√4的对数﹚

先算前面﹙lg5﹚²＋lg2×lg50部分
=lg5*lg5+lg2*(lg5+lg10)
=lg5*lg5+lg2*lg5+lg2
=lg5(lg5+lg2)+lg2
=lg5+lg2
=1
再算后面部分﹙以5为底√2的对数•以7为底√9的对数﹚／﹙以5为底1/3的对数•以7为底³√4的对数﹚
=(log5(√2)*log7(√9))/(log5(1/3)*log7(³√4))
=((lg√2/lg5)*(lg√9/lg7))/((lg(1/3)/lg5)*(lg³√4/lg7))
=(lg√2*lg√9)/(lg(1/3)*lg³√4)
=(1/2lg2*lg3)/(-lg3*(2/3)lg2)
=(1/2)/(-2/3)
=-3/4
所以答案是1-3/4=1/4