Given Information:
- The first term of the geometric progression (GP) is 2.
- The last term of the GP is 512.
- The common ratio of the GP is 2.

To Find:
The fifth term from the end of the GP.

Solution:

To find the fifth term from the end of the GP, we need to find the value of the fifth term counting backwards from the last term.

Step 1: Find the common ratio (r).
The common ratio (r) can be found by dividing the second term by the first term in the GP.
In this case, the first term is 2 and the second term can be found by multiplying the first term by the common ratio (r).
Let's assume the second term is a.
So, a = 2 * r

Step 2: Find the value of r.
The last term of the GP is given as 512.
We can find the value of r by dividing the last term by the second term.
512 = a * r
Substituting the value of a from Step 1, we get:
512 = (2 * r) * r
512 = 2r^2

Step 3: Solve for r.
Divide both sides of the equation by 2:
256 = r^2
Taking the square root of both sides, we get:
r = √256
r = 16

Step 4: Find the fifth term from the end.
To find the fifth term from the end, we can use the formula for the nth term of a geometric progression:
nth term = a * r^(n-1)

In this case, the last term is 512, and we want to find the fifth term from the end.
Using the formula, we have:
512 = 2 * 16^(n-1)

Simplifying the equation, we get:
256 = 16^(n-1)

Taking the logarithm of both sides, we get:
log(256) = log(16^(n-1))
log(256) = (n-1) * log(16)

Using the logarithmic property, we can rewrite log(16) as:
log(16) = log(2^4)
log(16) = 4 * log(2)

Substituting this back into the equation, we have:
log(256) = (n-1) * 4 * log(2)

Now, solve for (n-1):
(n-1) = log(256) / (4 * log(2))

Using a calculator, we can find the value of (n-1) to be approximately 3.5.

Step 5: Find the fifth term.
To find the fifth term, we need to add 5 to (n-1):
n = 3.5 + 5
n = 8.5

Therefore, the fifth term from the end of the geometric progression is the term with index 8.5.

Conclusion:
The fifth term from the end of the geometric progression is the term with index 8.5.