How to Convert a Complex Logarithm to a Complex Exponential

[CalcExplorer]

Okay, so I'm working with a rather frustrating problem with a calculus equation. I'm trying to solve a calculus equation which I conceptualized from existing methods involving complex number fractal equations. I'm very familiar with pre-calculus, while being self-taught in portions of calculus for practical applications in coding and higher dimensional mathematics, so bare with me on this.

Here's the premise:
I'm using the Mandelbrot Equation [ z = z² + i C ] and the Hausdorff Dimension [ N = s^d ], where d = ln(N)/ln(s), to create a new iterational equation, which fractally conforms to an already defined Hausdorff Dimension [ log(20)/log(2+φ) ], where φ = ((√5)+1)/2. _{This specific Hausdorff Dimension is of a Dodecahedron Fractal Flake.}

This provides the new equation:
z = z^{log(20)/log(2+φ)} + i C

I'm attempting to solve the first iteration of this new equation in quadratic form, the format the mandelbrot is solved in [ z₂ = a²-b² + 2abi ], as it's the format most advantageous for the graphical mapping of the fractal structure.

From what I've surmised thus far in order to solve such an equation I need to first convert the complex logarithm into a complex exponential using Euler's Formula, and then solve the new formula algebraically to derive the first iterational solution.

There seems to be a basis for this method, albeit with certain conditions in the solutions, and similar questions put up on this forum before, but I don't quite understand the principles enough to solve it myself.

These are the relevant mathematical references on the topic I've been able to find -
http://math.gmu.edu/~rsachs/m114/eulerformula.pdf
https://www.physicsforums.com/threads/eulers-formula-and-complex-logarithms-relationship.559665/
https://www.reddit.com/r/askscience/comments/2e3jnv/logarithms_of_complex_numbers_logarithms_with/

But that's pretty much it and I don't quite grasp how it's converted, since there's no direct examples of this process that I could find with conversions using Euler's Formula and Complex Logarithms.

Is this the proper way to go about solving such a problem?
And is there someone here that can help to find the solution?

 

[stevendaryl]

I'm not sure I understand what it is that you're doing, or what it is that you're asking. The number $log(20)/log(2+\varphi)$ is just a number, which you can calculate. Let's call it $k$. Are you just asking how to compute $z^k + iC$?

I don't know immediately what the relationship is between the fractal dimension of the iteration and the power $k$. I don't think the two are necessarily connected.

 

[CalcExplorer]

I'm not sure I understand what it is that you're doing, or what it is that you're asking. The number $log(20)/log(2+\varphi)$ is just a number, which you can calculate. Let's call it $k$. Are you just asking how to compute $z^k + iC$?

Yes, I'm aware that it's a calculable number, k = 2.3296217161703454689697018540751, but that's not the ideal format to compute $z^k + iC$. As I'm trying to define it's first solvable iteration in quadratic form.

I don't know immediately what the relationship is between the fractal dimension of the iteration and the power $k$. I don't think the two are necessarily connected.

This is a problem which was published before in a very interesting thesis, which also describes it's relationship between the fractal dimension of the iterations and the power $k$ :
https://www.math.hmc.edu/seniorthesis/archives/2003/shaas/shaas-2003-thesis.pdf

 

[Mark44]

Yes, I'm aware that it's a calculable number, k = 2.3296217161703454689697018540751, but that's not the ideal format to compute $z^k + iC$. As I'm trying to define it's first solvable iteration in quadratic form.

If the above is $z^k + iC$, there's no way that taking the log would be helpful, as there is no property that can break up $\log(A + B)$.
Or did you mean to write $z^{k + iC}$?
In that case $\log(z^{k + iC}) = (k + iC)\log(z)$.

This is a problem which was published before in a very interesting thesis, which also describes it's relationship between the fractal dimension of the iterations and the power $k$ :
https://www.math.hmc.edu/seniorthesis/archives/2003/shaas/shaas-2003-thesis.pdf

 

[CalcExplorer]

If the above is $z^k + iC$, there's no way that taking the log would be helpful, as there is no property that can break up $\log(A + B)$.
Or did you mean to write $z^{k + iC}$?
In that case $\log(z^{k + iC}) = (k + iC)\log(z)$.

Sorry, the confusion is from an error in the OP.

Given the aformentioned thesis the Hausdorff dimension can be expressed in polynomial form.
So, the new equation becomes:

$z=z^k+iC$

where, $k=z^k+iC$ [solved numerically giving the complex number julia set]
k = z^{(log(20)/log(2+φ))}$+ 2.329621i$

This means k is a complex logarithm able to be converted into a complex exponential using Eulers formula. Once converted it could be solved but for the z iteration.

At least in theory anyway, using this method would the conversion from a complex logarithm to a complex exponential be valid and possible? If it can be converted, what would be the solution for the quadratic form solution of $z=z^k+iC$?

 

[CalcExplorer]

Following the iteration therefore for k:
$k = k$₀^{(log(20)/log(2+φ))}$+2.329621i$
$k$₀$= 0 →k$₁$= 0^k + c ≡c$
$c = 2.329621i$
$k$₂$= c$^{(log(20)/log(2+φ))}$+2.329621i$

Which means:
$k = c$^{(log(20)/log(2+φ))}$+2.329621i$

After the conversion of the complex log into a complex exponential the new form of the equation
$k=c$^{(log(20)/log(2+φ))}
is solved in quadratic form, and then used in the $z=z^k+iC$ also solved in quadratic form