Let 'L' denotes the antilog of 0.4 to the base 1024 and 'M' denotes the number of digits in 6₁₀ (Given log, 02-03) and 'N' denotes the number of positive integers which have the characteristic 2, when the base of the logarithm is 6. Find the value of LMN.

To solve for the value of LMN, we need to find the values of L, M, and N step by step.

Finding L

Let L be the antilog of 0.4 to the base 1024. This can be expressed as:

L = 1024^0.4

Calculating this, we first recognize that 1024 is equal to 2^10. Therefore:

L = (2^10)^0.4 = 2^(10 * 0.4) = 2^4 = 16

Calculating M

M represents the number of digits in the number 6₁₀. The formula to find the number of digits in a number n in base b is:

Number of digits = ⌊log_b(n)⌋ + 1

For 6 in base 10:

M = ⌊log₁₀(6)⌋ + 1

Using a calculator, we find:

log₁₀(6) ≈ 0.7781

Thus:

M = ⌊0.7781⌋ + 1 = 0 + 1 = 1

Determining N

N is the number of positive integers with a characteristic of 2 when the base of the logarithm is 6. The characteristic of a logarithm is the integer part of the logarithm.

We need to find integers x such that:

log₆(x) ≥ 2

This implies:

x ≥ 6² = 36

Now, we consider the positive integers starting from 36. The integers are 36, 37, 38, ..., up to 6³ (which is 216):

6³ = 216

The count of integers from 36 to 215 is:

N = 215 - 36 + 1 = 180

Calculating LMN

Now that we have L, M, and N:

L = 16, M = 1, N = 180

Thus:

LMN = 16 * 1 * 180 = 2880

The final answer is:

LMN = 2880