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If P is the number of natural numbers wh...

If `P` is the number of natural numbers whose logarithms to the base 10 have the characteristic `pa n dQ` is the number of natural numbers logarithms of whose reciprocals to the base 10 have the characteristic `-q` , then find the value of `log_(10)P-(log)_(10)Qdot`

Text Solution

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`10^(p) leP lt 10^(p+1)`
` rArr P=10^(p+1) - 10^(p)`
` rArr P=9 xx 10^(p)`
Similarly, `10^(q-1)lt Q le10^(q)`
`rArr Q = 10^(q)-10^(q-1) = 10^(q-1)(10-1)=9 xx 10^(q-1)`
` :. log_(10)P-log_(10)Q = log_(10)(P//Q)`
` log_(10)10^(p-q+1)`
` p-q+1`
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