Question

Let $N$ be the number of integers whose logarithms to the base 10 have the characteristic 5 , and M the number of integers the logarithms to the base 10 of whose reciprocals have the characteristic 4. Find $\left(\log _{10} N-\log _{10} M\right)$.

Answer 1

We know that logarithms of all numbers lying between $10^2$ and $10^3$ will have characteristic2.
$\therefore N =10^6-10^5=9 \cdot 10^5$ ....(1)
Number of all integer the logarithm of whose reciprocals have the characteristic $(-2)$ are
$100,99, \ldots \ldots \ldots \ldots . . . . . .11$
i.e. $ 10^2-10$
M the number of integers the logarithms to the base 10 of whose reciprocals have the characteristic -4
$\therefore M =10^4-10^3=9 \cdot 10^3$ ......(2)
Hence, $\left(\log _{10} N -\log _{10} M \right)=\log _{10}\left(\frac{ N }{ M }\right)=\log _{10}\left(\frac{9 \cdot 10^5}{9 \cdot 10^3}\right)=\log _{10}(100)=2$.