Hi guys... I'm probably missing something pretty basic here but I can't seem to figure this out. I was working on a problem recently: for the complex functions f(z)=e(adsbygoogle = window.adsbygoogle || []).push({}); ^{z}and g(z)=z, find their intersections. This post is not about the problem, it is about something I noticed while tackling it (incorrectly).

Anyways, here's what I noticed: If you set these functions equal to each other, you get

e^{z}=z

So, naturally:

z=ln(z)

From here I saw that a basic substitution was applicable, so the equation can be rewritten:

e^{z}=ln(z)

Basically, what I have shown is that the function h(z)=e^{z}-z has the same zeroes as the function i(z)=e^{z}-ln(z). Now here's what's troubling me: what if the original problem that I gave you was to find the zeroes of i(z)? Originally, we obtained i(z) from h(z) by using a substitution, but is there some way that we can go in reverse from i(z) to h(z) using a "reverse substitution"? I'm sorry if this is rather unclear. Is there anything fundamental that I am missing?

Thanks a lot

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# Reversing substitution

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