What is the log of the beta function, how can it be simplified?

Question

$$B(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1} \, dt$$ is the beta function. What is the meaning and simplification of the log of the beta function:

$$\log\left(\int_0^1 t^{x-1}(1-t)^{y-1} \, dt\right) = \, ?$$

Answer 1

well: $$B(x,y)=\frac{\Gamma(x)\Gamma(y)}{\Gamma(x+y)}$$ so: $$\ln(B(x,y))=\ln(\Gamma(x))+\ln(\Gamma(y))-\ln(\Gamma(x+y))$$ then you can just use this instead??

Also there are identities relating to approximating $\ln(\Gamma(x))$ or finding its fourier series

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Also notice that: $$\ln\left(\prod a_i\right)=\sum\ln(a_i)$$