Question

If $P$ is the number of natural numbers whose logarithm to the base $10$ have the characteristic $p$ and $Q$ is the number of natural numbers logarithm of whose reciprocals to the base $10$ have the characteristic $-q$, then find the value of ${\log }_{10}P-{\log }_{10}Q$.

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

Answer 1

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

Let $x$ and $y$ be the numbers whose logarithm to the base $10$ have the characteristic $p$ and $q$ respectively.

${10}^p\le x<{10}^{p+1}\Rightarrow P={10}^{p+1}-{10}^p\Rightarrow P=9\times {10}^p$
Similarly, ${10}^{q-1}<y\le {10}^q$
$\Rightarrow$ $Q={10}^q-{10}^{q-1}={10}^{q-1}(10-1)=9\times {10}^{q-1}$
$\therefore$ ${\log }_{10}P-{\log }_{10}Q={\log }_{10}\left(P/Q\right)={\log }_{10}{10}^{p-q+1}=p-q+1$