Probability of [log₁₀(4x)]-[log₁₀(x)]=0 for x (0,1)

[Jameson]

Let \(\displaystyle x\) be chosen at random from the interval \(\displaystyle (0,1)\). What is the probability that \(\displaystyle [\log_{10}(4x)]-[\log_{10}(x)]=0\)?

Here \(\displaystyle [x]\) denotes the greatest integer that is less than or equal to \(\displaystyle x\).

This is multiple choice, but I don't think posting the possibilities are necessary. I would give my input to the problem of course, but I have no idea where to start.

 

[shmoe]

First thing to do, for a given integer n work out the values of x where [log_10(x)]=n.

By the way, you want "tex" in the tags instead of "math".

 

[Jameson]

Thanks, and I noticed by mixplaced tags as well. I copied this from my post at a site where the \(\displaystyle tags are used instead of [ tex ] ones. I can't edit it now though, so if a moderator could that'd be nice.\)