1除以以0.4为底4的对数是否等于以4为底0.4的对数?怎么算到的,thanks

Question

1除以 以0.4为底4的对数是否等于以4为底0.4的对数?
怎么算到的,thanks

Answer 1

1/(log 0.4 (4))
=1/(log 0.4 (0.4*10))
=1/(log 0.4 (0.4)+log 0.4 (10))
=1/(1+log 0.4 (10))
=1/(1+lg(10)/lg(0.4))
=1/(1+1/lg(4/10))
=1/(1+1/(lg(4)-lg(10)))
=1/(1+1/(lg(4)-1))
=1/((lg(4)-1)+1)/(lg(4)-1))
=(lg(4)-1))/(lg(4))
=1-1/(lg(4))
log 4 (0.4)
=log 4 (4/10)
=log 4 (4)-log 4 (10)
=1-log 4 (10)
=1-lg(10)/(lg(4))
=1-1/(lg(4))
所以：1/(log 0.4 (4))=log 4 (0.4)