Puzzle Time - find all roots for (x^2-7x+11)^(x^2-13x+42) - 1

[Axtremus]

Find all the roots for (x^2-7x+11)^(x^2-13x+42) - 1
I.e., find all possible values for x that satisfy the condition (x^2-7x+11)^(x^2-13x+42) = 1

The ^ symbol in this case means "raised to the power of", as in x^y means "x raised to the power of y".

This problem is somewhat peculiar in that most computer programs seem to have problem with it (e.g., try to plot it with the Grapher app that ships with macOS or try to plot it with Excel and you will see that there is a section of the graph "missing", like the computer does not know how to plot that section of the graph), but it can be completely solved by humans analytically.

[Klaus]

The 4 obvious solutions are when the base is 1 or the exponent is 0.

But from what I can see there are at least 6 solutions; two more solutions arise when the base is -1 and the exponent an even number.

Are you only looking for "real" solutions or do complex numbers count, too? (your function is only a partial function on the reals)

[jon-nyc]

||x=6,7

(x^2-7x+11)^(x^2-13x+42) = 1

(x-6)(x-7)* ln(x^2-7x+11) =0

ln(...) never = 0, so the expression is 0 only when the first two expressions equal 0||

[Klaus]

||2,3,4,5,6,7||

I assume there are more (complex number) solutions, though.

[jon-nyc]

||I see 2 and 5 also get the expression (x^2-7x+11) to equal 1 which also satisfies the equation.

So x e (2,5,6,7)||

[Klaus]

@Klaus said in Puzzle Time - find all roots for (x^2-7x+11)^(x^2-13x+42) - 1:

I assume there are more (complex number) solutions, though.

Yes, as I assumed there are indeed more solutions in the complex numbers, but they don't look as nice. For instance, here's another solution:

2.27245569233 + 0.94176503421i

[Axtremus]

@Klaus gave the complete set of real roots, great job!

I have not sort through complex roots, so not sure how many of them there are. Cool to see one though!