I was wondering how one would multiply two logarithms together?
Say, for example, that I had:
$$\log x·\log 2x < 0$$
How would one solve this? And if it weren't possible, what would its domain be?
Thank you!
(I've uselessly tried to sum the logs together but that obviously wouldn't work. Was just curious as to what it would give me)
EDIT: Thank you everyone for the answers!
Best Answer
As $\log2x=\log2+\log x,$
$$0>\log x\cdot\log2x=\log x(\log x+\log 2)$$
Now if $(y-a)(y-b)<0$ with $a<b,$ we can prove $$a<y<b$$
So, here we have $$-\log 2<\log x<0$$
$$2^{-1}<x<1$$