How can I determine the roots of (x)*(1/9)^(1/9)^x - 1 = y using DESMOS?

[laplacianZero]

[(x)*(1/9)^(1/9)^x ] - 1 = y

How do you find the roots?

 

[jedishrfu]

Have you tried plugging in numbers like 0,1,2...?

Have you tried to plot it to see what you're dealing with?

This site can plot functions:

https://www.desmos.com/calculator

 

[mfb]

If you cannot find analytic solutions, things are easier. plot it, calculate approximations for the root, and you are done.

 

[laplacianZero]

Besides plotting to find approximate root or using the Newton raphson method, are there any other ways?

 

[Mark44]

Besides plotting to find approximate root or using the Newton raphson method, are there any other ways?

Since x appears as both the base and as an exponent, the only non-graphical, non-numerical alternative is the Lambert W function. See https://en.wikipedia.org/wiki/Lambert_W_function

 

[jedishrfu]

The desmos graph calculator shows two zeros at ? and at ? where $0 < x < 10$

The ? are left to the student.

 

[mfb]

The desmos graph calculator shows two zeros at ? and at ? where $0 < x < 10$

The ? are left to the student.

The expression does not have a second zero, at least not with the conventional interpretation of a^b^c as a^(b^c).

 

[jedishrfu]

Yes, you're right. I can't reproduce what I typed into the Desmos calculator.

Last night it gave me a curve that looked something like the Lambert W curves and crossed at x=1.411 and x=9.

Something like this:

$y = \left(x\right)\cdot e^{-x}\ -\frac{1}{5}$

However, now when I plot it, I see only x=1.79.

$y = x\cdot \left(\frac{1}{\left(9\right)}\right)^{\left(\frac{1}{\left(9\right)}\right)^x}-1$

 

[mfb]

You get that if you interpret a^b^c as (a^b)^c = a^(b*c).

 

[jedishrfu]

The DESMOS graphing calculator input editor is a little wonky. Its different in that while it completes parentheses when you type the closing parentheses it add another on instead saying okay got that.

To get around this behavior, you have to instead tab out of the closing parentheses to get where you want to be.

 

[jedishrfu]

You get that if you interpret a^b^c as (a^b)^c = a^(b*c).

Yes, that may have what happened and I didn't notice. I should have saved the expression that was input then I didn't where I went wrong.

I do know earlier I had thought the x*(1/9) factors were x^(1/9) instead which produced yet a different result.

One nice thing about the DESMOS is that as you type in the expression and it looks correct you can copy and paste it here as its valid Latex.