# Find the number of positive integers such that logarithm of whose reciprocals to the base 10 has the characteristic $-2$.

Find the number of positive integers such that logarithm of whose reciprocals to the base 10 has the characteristic $-2$.

Let $x$ be a positive integer.
Now the characteristic of $\log_{10}(\frac{1}{x})$ is $-2$
I dont know how to solve further.How to count number of positive integers?Please help me.Thanks.

• I don't think you have the right definition of characteristic. You need a floor or ceiling function somewhere. – Ted Nov 30 '15 at 3:03

If that is your question, $log_{10} {1/x} = -2$

${1/x} = 10^{-2}$

$x = 100$

does this help?

• No,this is not the answer.The answer is $90.$ – Vinod Kumar Punia Nov 30 '15 at 2:34
• @Vinod, can u explain how u got the answer. – NiroshaR Nov 30 '15 at 2:35
• I have just quoted the book's answer,i dont know how it came. – Vinod Kumar Punia Nov 30 '15 at 2:38
• @vinod does the book meant number of such positive integers is $90$ – Ekaveera Kumar Sharma Nov 30 '15 at 6:35
• Yes,number of such positive integers is $90$,@EkaveeraKumarSharma – Vinod Kumar Punia Nov 30 '15 at 6:40