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 ( log 10 N- log 10 M).

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 ( log 10 N- log 10 M). (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

We know that logarithms of all numbers lying between

1 0^{2}

and

1 0^{3}

will have characteristic2.

∴ N = 1 0^{6} - 1 0^{5} = 9 \cdot 1 0^{5}

....(1)
Number of all integer the logarithm of whose reciprocals have the characteristic

(- 2)

are

100, 99, \dots\dots\dots\dots ......11

i.e.

1 0^{2} - 10

M the number of integers the logarithms to the base 10 of whose reciprocals have the characteristic -4

∴ M = 1 0^{4} - 1 0^{3} = 9 \cdot 1 0^{3}

......(2)
Hence,

(lo g_{10} N - lo g_{10} M) = lo g_{10} (\frac{N}{M}) = lo g_{10} (\frac{9 \cdot 1 0 ^{5}}{9 \cdot 1 0 ^{3}}) = lo g_{10} (100) = 2

.

Q. 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 (log10​N−log10​M).

Q. 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 $(lo g_{10} N - lo g_{10} M)$ .