How can you take the summation of log(2^x-1)

Question

I want to multiply $\frac{1}{2} * \frac{3}{4} * \frac{7}{8}$ ... etc for any arbitrary integer. The problem is that I can't seem to simplify it any further than

$\frac{\exp\left(\sum_{n=1}^{x}\ln\left(2^{n}-1\right)\right)}{2^{\frac{x^2+x}{2}}}$

What I've learned is that the integral $e^{\int_{0}^{x}\ln\left(2^{t}-1\right)\ dt}$ approximates the problem very well but I just can't get farther than that. Not even solvers will put out what that top term is. Is this one of those things where it can't be solved algebraically even though there is a solution?

Does this answer your question? What's the limit of $\prod_1^\infty \left(1-\frac{1}{2^n}\right)=(1-1/2)(1-1/4)(1-1/8)...$? — Manifoldski, Oct 10 '22 at 22:29
@Manifoldski: that question is for the infinite product while this one is for a finite number of terms — Ross Millikan, Oct 10 '22 at 22:32
The difference between $\sum_{n=1}^x\ln(2^n-1)$ and $\int_0^x\ln(2^t-1)dt$ appears to grow linearly with $x$, which would mean you have an exponentially bad approximation. — anon, Oct 10 '22 at 22:34
I think an examination of the properties of the polynomials $P_m(z) = \prod_{n=1}^m (1-z^n)$ would be more helpful for your cause than logarithms and exponentials. — Paul Sinclair, Oct 11 '22 at 16:31
I'm not after the number it approaches, I'm looking for an algebraic function that outputs this series https://oeis.org/A005329 at every integer. — Logos King, Oct 15 '22 at 22:26
Technically, neither $\exp$ nor $\ln$ are "algebraic functions," they are transcendental. An algebraic function is one which satisfies some polynomial equation, like $\sqrt{x}$ is the root of $u^2-x$ (as a polynomial in $u$, treating $x$ as a parameter). I assume you mean you want a closed form involving so-called elementary functions. I don't think such a form exists. — anon, Oct 16 '22 at 17:16

How can you take the summation of log(2^x-1)

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