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I propose :

var('a,b,x,x1,x2,x4')
eq=e^(-a*log(x1)/(a+b)+a*log(x1)/(a+b)+a*log(x2)/(a+b)-a*log(x4)/(a+b))
show(eq.simplify_log())

I propose :

var('a,b,x,x1,x2,x4')
eq=e^(-a*log(x1)/(a+b)+a*log(x1)/(a+b)+a*log(x2)/(a+b)-a*log(x4)/(a+b))
show(eq.simplify_log())

SageMath doc on symbolic expression

but I admit that me too , I am a bit lost in all these different simplifying functions!