Question 228323
Let x = a positive number.


Then -x = a negative number.


log(b) < 0 implies that log(b) = -x because -x < 0.


Now log(b) = -x if and only if 10^(-x) = b.


This is by basic definition of logarithms.


10^-x is the same as 1/10^x by definition.


Since x >= 0 by definition, then the smallest 10^x could be would be 1 because 10^0 = 1. 


Any other value of x > 0 would result in 10^x being greater than 1.


Example:


10^0.1 = 1.2589.....

10^0.00001 = 1.0000023026


Bottom Line is the smallest 10^x can be is 1.


Now, if 1/10^x = b, this means that the largest b can be is 1 because 1/1 = 1.


so, to answer your question:


If log(b) < 0, this means that 0 < b < 1


Some examples:


log(2) = .3...
log(1) = 0
log(.9) = -.04...
log(.5) = -.301...
log(.1) = -1
log(0) = Error - can only take log of a number > 0


So that's the answer to your question.


log(b) < 0 if and only if b is greater than 0 and b is smaller than 1.