Question 228323: Please help!
If log b < 0, what can you say about b?
Thanks so much!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! 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.
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