To simplify this expression, we can use the properties of logarithms:

Logarithmic Property: log_b(xy) = log_b(x) + log_b(y)
Logarithmic Property: log_b(x^a) = a * log_b(x)

Applying these properties,

log2 3 + log2 24 - 4 log4 3
= log2 (3 24) - log4 3^4
= log2 72 - log4 81
= log2 (2^3 3^2) - log4 (3^4)
= log2 2^3 + log2 3^2 - 4 log2 3
= 3 log2 2 + 2 log2 3 - 4 log2 3
= 3 + 2 * log2 3 - 4 log2 3
= 3 - 2 log2 3

Therefore, log2 3 + log2 24 - 4 log4 3 simplifies to 3 - 2 log2 3.