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Question

If P is the number of natural numbers whose logarithms to the base 10 have the characteristic p and Q is the number of natural numbers, logarithms of whose reciprocals to the base 10 have the characteristic q, then the value of log10Plog10Q is

A
pq
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B
pq+1
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C
p+q1
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D
p+q
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Solution

The correct option is B pq+1
Let M be the natural numbers whose logarithms to the base 10 have the characteristic p
Then 10pM<10p+1
Number of natural numbers satisfying the above inequality is,
P=10p+110p=9×10p

Let N be the natural numbers logarithms of whose reciprocals to the base 10 have the characteristic q
Then 10q1N<10q+1
10q1<N10q
Number of natural numbers satisfying the above inequality is,
Q=10q10q1=9×10q1

log10Plog10Q
=log10PQ
=log109×10p9×10q1
=log1010pq+1
=pq+1

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