To solve the equation 3:2^x-1=24-[4^2-(2^2-1)], we need to simplify both sides of the equation and then solve for x.

First, let's simplify the right side of the equation:

24 - [4^2 - (2^2 - 1)]
24 - [16 - (4 - 1)]
24 - [16 - 3]
24 - 13
11

Now, let's rewrite the equation with the simplified right side:

3:2^x-1 = 11

To solve for x, we need to isolate the exponential term on the left side. We can do this by multiplying both sides of the equation by 2^x-1:

(2^x-1)(3:2^x-1) = (2^x-1)(11)

On the left side, the (2^x-1) terms cancel out, leaving us with:

3 = 11(2^x-1)

Now, let's distribute the 11 on the right side:

3 = 22^x - 11

Next, let's move the constant term to the other side of the equation:

22^x = 3 + 11
22^x = 14

To solve for x, we need to take the logarithm of both sides. Let's take the logarithm base 2:

log2(22^x) = log2(14)

Using the logarithmic property logb(a^c) = c*logb(a), we can simplify the left side:

x*log2(22) = log2(14)

Now, divide both sides of the equation by log2(22):

x = log2(14) / log2(22)

Using a calculator, we can find the approximate value of x:

x ≈ 0.847

Therefore, the solution to the equation 3:2^x-1=24-[4^2-(2^2-1)] is x ≈ 0.847.