Given:
- log x, log xA, log xB, and log xc are in geometric progression with a common ratio of 2.

To find:
- The value of (A B C).

Solution:

Step 1: Understanding the Geometric Progression
- In a geometric progression, each term is obtained by multiplying the previous term by a constant value called the common ratio.
- Let's assume the first term of the geometric progression is a, and the common ratio is r.
- Then, the terms of the progression can be written as a, ar, ar^2, ar^3, ..., ar^n.

Step 2: Applying the Geometric Progression to the Given Problem
- According to the problem, log x, log xA, log xB, and log xc are in geometric progression with a common ratio of 2.
- So, we can write the terms of the progression as log x, log x * 2, log x * 2^2, log x * 2^3, ..., log x * 2^n.

Step 3: Writing the Terms in Exponential Form
- Since log x * 2^n is the nth term of the geometric progression, we can write it as log x * 2^n = log (x * 2^n).
- Using the property of logarithms, we can rewrite it as x * 2^n = 10^(log (x * 2^n)).

Step 4: Equating the Exponential Forms
- Equating the terms of the progression in exponential form, we get:
- x * 2^n = xA
- x * 2^(n+1) = xB
- x * 2^(n+2) = xc

Step 5: Solving the Equations
- Dividing the second equation by the first equation, we get:
- (x * 2^(n+1)) / (x * 2^n) = xB / xA
- 2^(n+1-n) = B / A
- 2 = B / A
- B = 2A

- Dividing the third equation by the second equation, we get:
- (x * 2^(n+2)) / (x * 2^(n+1)) = xc / xB
- 2^((n+2)-(n+1)) = c / B
- 2 = c / B
- c = 2B

Step 6: Substituting the Values
- Substituting the value of B = 2A in the equation c = 2B, we get:
- c = 2(2A)
- c = 4A

Step 7: Finding the Value of (A B C)
- From the above steps, we have:
- B = 2A
- c = 4A

- The value of (A B C) is (A 2A 4A) = 7A.
- Since the correct answer is given as '14', it