Let's Solve An Eeezee (e, z) Exponential Equation

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[@uwelinzbauer3973]

👍thanks!

[@SyberMath]

No problem!

[@andrewcoxon516]

How do we feel about the necessity of rationalizing denominators? I generally tell my students that it's archaic and that I really just want them to know it for testing purposes. Your comment about the 'reciprocal function' was particularly interesting. What if instead of rationalizing 1/z, we just express it as z^-1 or even some abstract function like rec(z)? I feel like the reciprocal of a single number, even a complex one, is more meaningful than the ratio of two numbers.

[@MrSeezero]

Thanks for demonstrating the polar form of a complex number.

[@Physics_HB]

I solved it mentally

[@mcwulf25]

Someone here will have solved it pedally.

[@scottleung9587]

Cool - I got the principal solution for n=0.

[@goldfing5898]

To get the real solutions:
e^(1/x) = -e
Divide by e:
e^(1/x) / e = -1
e^(1/x) / e^1 = -1
e^(1/x - 1) = -1
Take the ln() on both sides:
1/x - 1 ÷ ln(-1)
Ooops, ln() is not defined for negative real numbers.
Thus no real solution :-)

[@bnice29]

Correct, there is no solution. For desmos,
let y = e^(1/x) & y = -e.
The graphs do not intersect, thus no solution.
2nd method, raise each side to the x power.
e = -e^x
Ln (e) = ln (-e)^x
1 = x ln(-e)
x = 1/(ln(-e))
Does not exist since you cannot take the natural log of a negative number.

[@SyberMath]

complex solutions?

[@lawrencejelsma8118]

A warning from one mathematics engineer to a liberal mathematics master is to not destroy the complex real and imaginary polar plane like all those talking about a same point mapped into an x from f(x) = √x (the square roots debate of do we include or not include both roots) accepted thoughts on primary roots. In Physics and the sciences while we are interested in harmonics we know the primary data point thoughts are adequate enough. That is, to understand more what the differences of mathematics in imaginary and fantasies of all 2nπ type angular equivalences we still have to, by science and real world applications, satisfy a primary mapping on the Complex polar plane. Physics and sciences do understand attenuations and gains or same valued harmonics in 3 dimensional helix space extending all angle 2πns to not map to a same data point in a third dimension. This makes the other 2πn math angles necessary with vector spaces. It's hard to see (since I am in the mathematics sciences) mathematics by liberal arts anything useful in non-helix or other dimensional vector space mathematicians viewpoints!! 😂. You are still a great math "every different solutions" math teacher I highly recommend. I don't see your math correlating to how engineering mathematicians have to think all the time but you are great in your illustrations and knowledge!! 👍👍

[@mcwulf25]

I don't like the root symbol being used for that reason. What does it even mean in the complex world? And what does the xth root mean, on the basis that square root needs +/- prefix to elicit two solutions.

[@SyberMath]

❤️

[@_ilsegugio_]

solving this cute equation you stumble into: e^1+iπ=-e
which is somewhat interesting.
another way to realize you cannot express zero via complex exponentiation
there's no such z∈ℂ by which:
e^z=-e^z