I am still confused with the problem Simply Exponential \[\quad \quad \quad \quad { x }^{ x }=x\\ \Longrightarrow x={ x }^{ 1/x }\\ \Longrightarrow { x }^{ x }={ x }^{ 1/x }\\ \Longrightarrow x\ln { x } =\frac { 1 }{ x } \ln { x } \\ \Longrightarrow \ln { x } (\frac { { x }^{ 2 }-1 }{ x } )=0\ \\ \Longrightarrow x=\pm 1(Correct \quad answer)\] Now,since we have included a step \(x\ln { x } =\frac { 1 }{ x } \ln { x }\) in our solution,according to the domain of the logarithmic function, x should be positive,thus omitting -1.Then why -1(as we cannot take log of a negative number)?

**Then, I am confused that the given expression(actually an equation) is a function or not.** I mean, if it is given that \(f\left( x \right) ={ x }^{ x }-x\) and then we are asked to find the roots or zeroes of the function, then the answer will be only +1 or still be both+1 & -1?(I think that it should be only +1 as the domain of \(x^{x}\) is \(x>0\) )

Also, I don't know why, but is the second step correct and safe?

Please clear my doubts,and please provide any kind of guidance wherever you think is needed. Thanks

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TopNewestYou are right that, if we just want to work with continuous functions over the field of real numbers, the domain of both \(x^{x}\) and \(\ln(x)\) is \(x \gt 0.\) So the step where we take the natural log of both sides can only be done under the condition that \(x \gt 0,\) and hence with this process only the solution \(x = 1\) is valid.

However, the function \(f(x) = x^{x}\) is also purely real, (i.e., has an imaginary component of \(0*i\)), whenever \(x\) is a negative integer, and since \((-1)^{-1} = \dfrac{1}{(-1)^{1}} = -1\) we also have that \(x = -1\) is a solution to the equation \(x^{x} = x.\)

So while \(g(x) = x^{x} - x\) is a continuous function only for \(x \gt 0,\) it does still have a "zero" at \(x = -1.\) – Brian Charlesworth · 1 year, 3 months ago

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function,then what will be it's(I mean the function's) domain? (As the zero sholud lie in it's domain).Will it's domain be (all positive real numbers + negative integers)? Also, according to you, -1 is not a solution which we get mathematically(I mean analytically) and it is just by random hit and trial, so if in a subjective paper, we have to conclude -1(it is a bit funny and pointless, but just...),what's the best way to do it? – Abhijeet Verma · 1 year, 3 months ago

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When dealing with powers of negative numbers it makes more sense to work in the field of complex numbers. That way we can expand the domain and such anomalies then become "analytic" solutions. – Brian Charlesworth · 1 year, 3 months ago

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– Calvin Lin Staff · 1 year, 3 months ago

Wolfram doesn't deal well with isolated point domains (even if you think that they are nice like the integers).Log in to reply

– Brian Charlesworth · 1 year, 3 months ago

So can we then legitimately choose that the domain of \(g(x) = x^{x} - x\) is the set of all positive reals as well as all negative integers? Then both \(1\) and \(-1\) would be valid roots. If we wanted to deal only with a continuous function then we could restrict the domain to "just" \(x \gt 0.\)Log in to reply

Of course, for a function to be completely well defined, the domain needs to be stated. In the event that the domain isn't stated, it is extremely open to interpretation. E.g. What are the zeros of \( f(x) = x^2 + 1 \)? Those who say no zeros (working in the domain of real numbers) are equally correct as those who say zero of \(i \) and \( -i \) (working in the domain of complex numbers).

In the event that the domain is over specified, then we simply ignore values where the function doesn't make sense. In this case, it means that we remove those negative non-integer values. – Calvin Lin Staff · 1 year, 3 months ago

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– Brian Charlesworth · 1 year, 3 months ago

O.k., great. Thanks for clearing that up. :)Log in to reply

See Ivan's solution, which avoids the numerous issues that you have raised.

Whenever we manipulate an equation, we have to be careful about the steps that we're taking, and check to see if we're introducing extraneous solutions, or losing actual solutions. For example, by taking logs, we need to assume that \( x > 0 \). – Calvin Lin Staff · 1 year, 3 months ago

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@Janardhanan Sivaramakrishnan Sir & @Brian Charlesworth Sir please help. – Abhijeet Verma · 1 year, 3 months ago

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Let \(x=r e^{i\theta}\). Then, \(\ln x = \ln r + i\theta\) (This step makes it possible to have natural logarithms for negative or even complex numbers).

\((\ln r + i\theta)(r e^{i\theta} - r^{-1} e^{-i\theta})=0\)

Case 1 : \(\ln r = 0\) and \(\theta = 0\). This implies \(x=1\)

Case 2 : \(r e^{i\theta} = r^{-1} e^{-i\theta}\)

This means \(r^2 e^{2i\theta} = 1\)

Which in turn means that \(|r| =1\) and \(2\theta = 2n\pi\), or \(\theta = 0,\pi\)

This gives \(x=1\) again. But, we also find the second solution \(x=-1\). – Janardhanan Sivaramakrishnan · 1 year, 3 months ago

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– Abhijeet Verma · 1 year, 3 months ago

Thanks Sir for clearing the confusion.Log in to reply

I think \( x = +1\) is the only correct solution because domain of our function is \( x > 0 \) – Akhil Bansal · 1 year, 3 months ago

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– Calvin Lin Staff · 1 year, 3 months ago

It is not stated that the domain is \( x > 0 \). It asks for "distinct real solutions".Log in to reply

Before finding the solution of some function, don't we first check its domain? – Akhil Bansal · 1 year, 3 months ago

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If, however, we were asked for the number of roots of the function \(f(x) = x^{x} - x,\) then we would have be mindful of the domain. If we establish the domain as \(x \gt 0\) then there would be only one valid root, namely \(x = 1.\) If we were to also include the set of all negative numbers in the domain then \(x = -1\) would also be a valid root. The more "standard" choice of domain would be \(x \gt 0,\) though, since \(f(x)\) would then be continuous and differentiable over all the domain. – Brian Charlesworth · 1 year, 3 months ago

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– Abhijeet Verma · 1 year, 3 months ago

Thanks for inserting the graph. Yes, analytically,only +1 is the solution or zero of the function(still, -1 is confusing as \(f(-1)=0\)),as mentioned by Brian Sir, but -1 is a valid solution of the equation \(x^{x}=x\) [if not considering it a function],which we get by hit & trial.Log in to reply

– Akhil Bansal · 1 year, 3 months ago

From where do you get this question?Log in to reply

– Abhijeet Verma · 1 year, 3 months ago

I have mentioned the link in the note itself.Log in to reply

– Akhil Bansal · 1 year, 3 months ago

There can be a big debate on this self-contradicting question.Log in to reply

– Abhijeet Verma · 1 year, 3 months ago

What surprised me was that the all the reports were marked as resolved.Log in to reply

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BTW, Sir, can you please go through my note and comments and tell if -1 will be a zero of the function or not. Thanks – Abhijeet Verma · 1 year, 3 months ago

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– Calvin Lin Staff · 1 year, 3 months ago

It is a zero of the function since \( f(-1)= 0 \).Log in to reply

– Abhijeet Verma · 1 year, 3 months ago

Thanks sir for clearing the confusion.Log in to reply