I am a genetics researcher and my math is quite dusty. I have a column of gene expression values that I need to transform with an equation, but I do not know how to solve it. So here goes:
I know that this equation is true: $$b!= (\log_{2}F) !$$ I have the value of $b$, which is a constant.
So how do I calculate $F$?
I suppose I cannot just remove the factorials on both sides? (because $0!=1!$ for example). Please help!
As JMoravitz comments, if $b$ is an integer greater than $1$ you can just remove the factorial signs, leading to $$b=\log_2 F\\F=2^b$$ The only case of two different integers with the same factorial is the one you cite, $0!=1!=1$ Even if $b$ is not an integer and you are using the $\Gamma$ function with $b!=\Gamma(b+1)$ for $b\gt 1$ you can just remove the factorials because the Gamma function is monotonically increasing for arguments greater than $2$.
Over the range $-2 \le b \le 2$ the gamma function is plotted below. The horizontal axis is $b$ and the vertical axis is $\Gamma(b+1)=b!$ Over most of the range there are two values of $\log_2 (F)$ that can give the same factorial as $b$. One of them is $F=2^b$. I don't know how to find the other except by numerical root finding. For example, if $b=1.5, b! \approx 1.32934$ Then $\log_2(F)$ can equal $-4.967, -4.111, -0.3191,$ or $1.5$. There will be more roots even more negative.