Question

If [log101] + [log102] + [log103] + [log104] + ...... + [log10n] = n,
where [x] denotes the greatest integer less than or equal to x, then
(1) 96 ≤ n < 104
(2) 104 ≤ n < 107
(3) 107 ≤ n < 111
(4) 111 ≤ n < 116

Answer 1

Answer:

the correct answer is (3) 107 ≤ n < 111

Step-by-step explanation:

log₁₀1 = 0

log₁₀10 = 1

log₁₀100 = 2

since [x] denotes the greatest integer less than or equal to x

=> [log₁₀2] = [log₁₀3] = .......... = [log₁₀9] = 0

=>[log₁₀1] + [log₁₀2] + [log₁₀3] + .......... + [log₁₀9] = 0

also,

[log₁₀10] = [log₁₀11] = ................ = [log₁₀99] = 1

=> [log₁₀10] + [log₁₀11] + ................ + [log₁₀99] = 90

also

[log₁₀100]  =  [log₁₀101] =  [log₁₀102] = .........................=  [log₁₀999] = 2

hence ,

[log₁₀1] +  [log₁₀2] .....+  [log₁₀9] +  [log₁₀10] + ...... + [log₁₀99] +  [log₁₀100] + ...... [log₁₀n] = n

=> 0 + 90 + [log₁₀100] + ...... [log₁₀n] = n

=> 90 + 2(n - 99) = n

=> 2n - n - 198 + 90

=> n = 108

Hence the correct answer is (3) 107 ≤ n < 111