To solve for \( x \) in the equations provided, we first need to calculate the value of \( A \):

\[
A = 2 + 2^2 + 2^3 + ... + 2^{100}
\]

This is a geometric series where the first term \( a = 2 \) and the common ratio \( r = 2 \). The number of terms is \( 100 \) (from \( 2^1 \) to \( 2^{100} \)), giving us:

\[
A = a \frac{r^n - 1}{r - 1}
\]

where \( n \) is the total number of terms.

Since we start counting terms from \( 2^1 \), we have \( n = 100 \). Thus, we can substitute:

\[
A = 2 \frac{2^{100} - 1}{2 - 1} = 2(2^{100} - 1) = 2^{101} - 2
\]

Now we have \( A = 2^{101} - 2 \).

Next, we need to solve the equation:

\[
2^{2x-1} - 2 = B
\]

Setting \( B \) equal to \( A \):

\[
2^{2x-1} - 2 = 2^{101} - 2
\]

Adding \( 2 \) to both sides:

\[
2^{2x-1} = 2^{101}
\]

Since both sides are powers of \( 2 \), we can set the exponents equal to each other:

\[
2x - 1 = 101
\]

Solving for \( x \):

\[
2x = 102 \\
x = 51
\]

Thus, the value of \( x \) is:

\[
\boxed{51}
\]