# Why MMA refuses to simplify the following simple expression?

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.