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 ${\log }_{10}P-{\log }_{10}Q$ is

• A. $p-q$
• B. $p-q+1$
• C. $p+q-1$
• D. $p+q$

Answer 1

The correct option is B $p-q+1$

Let $M$ be the natural numbers whose logarithms to the base $10$ have the characteristic $p$
Then ${10}^p\le M<{10}^{p+1}$
Number of natural numbers satisfying the above inequality is,
$P={10}^{p+1}-{10}^p=9\times {10}^p$

Let $N$ be the natural numbers logarithms of whose reciprocals to the base $10$ have the characteristic $-q$
Then ${10}^{-q}\le \frac{1}{N}<{10}^{-q+1}$
$\Rightarrow {10}^{q-1}<N\le {10}^q$
Number of natural numbers satisfying the above inequality is,
$Q={10}^q-{10}^{q-1}=9\times {10}^{q-1}$

$\therefore {\log }_{10}P-{\log }_{10}Q$
$={\log }_{10}\frac{P}{Q}$
$={\log }_{10}\frac{9\times {10}^p}{9\times {10}^{q-1}}$
$={\log }_{10}{10}^{p-q+1}$
$=p-q+1$