I have the following simple expression
Exp[(Log[1 + x]^2 - Log[x]^2)/Log[1 + 1/x]] - x^2
that can be shown to simplify to x
for positive x
(and same for $x < -1$ if principal branch of log
is taken, $\log (x) = i \pi + \log (-x)$). However, none of the following
Simplify[Exp[(Log[1 + x]^2 - Log[x]^2)/Log[1 + 1/x]] - x^2]
FullSimplify[Exp[(Log[1 + x]^2 - Log[x]^2)/Log[1 + 1/x]] - x^2]
Simplify[Exp[(Log[1 + x]^2 - Log[x]^2)/Log[1 + 1/x]] - x^2,
Assumptions -> {x > 0}]
FullSimplify[Exp[(Log[1 + x]^2 - Log[x]^2)/Log[1 + 1/x]] - x^2,
Assumptions -> {x > 0}]
finds this simplification. Why is that? To show that the expression simplifies accordingly, use $a^2 - b^2 = (a+b)(a-b)$ formula inside the exponential and in the denominator $\log (1+1/x) = \log (1+x) - \log(x)$. Some stuff cancels out and you'll be left with $\exp \log (x+x^2) - x^2 = x$. I have no clue why MMA didn't choose to go down this route, I assumed that the common formulas like $a^n - b^n$ are known to it. I tried Simplify[Log[1 + 1/x] + Log[x], Assumptions -> {x > 0}]
and it correctly returned log[1+x]
, so this might not be the issue here.
This specific expression came up in some wider context, I frequently use MMA to simplify expressions after integration to get some nicer form and was surprised that this couldn't be simplified.
Maple
and it can do it. $\endgroup$Simplify
at leastReduce[Exp[(Log[1+x]^2-Log[x]^2)/Log[1+1/x]]-x^2==x && x>0,x]
returns0<x<Infinity
but it says it cannot accomplish this when givenx < -1
which I am guessing is because of the principal branch issue. $\endgroup$PowerExpand[Factor[#]]&//@ex
seems to work, whereex
is your expression. $\endgroup$