I have a problem where I need to know what J is. I do x^J and get y. For example, if I do 5^J, I would want to get 55 as y. Same with 4^J = 30. When J is 2.455, it works up to 4 only! I need for decimals of J! Is there an equation or something so I can make a program to calculate J?
I want to see if It is possible to get J to work with everything. For example if J = 2.45568392948... And works with 4^J, 5^J etc. To give the right answers for square pyramid numbers. Like, 4 = 30, 5 = 55 etc. So I don't need a new J value for each.
Let me know if I was unclear! It's a confusing topic :D
If you know $x,y$ and want $J$, you can take the log of both sides. $$x^J=y\\ J \log x = \log y \\ J=\frac {\log y}{\log x}$$
Use any base of logs that you like. Looking at your examples, though, $5^{2.455} \approx 51.9962, 5^{2.456} \approx 52.0799$, so I don't know how you get $55$. Also $4^{2.455} \gt 30$ but just a bit
Added after the edit about square pyramidal numbers: You will not be able to find a single value of $J$ that works for any set of figurate numbers because they are expressed by a polynomial. The square pyramid numbers are $F(n)=\frac 16n(n+1)(2n+1)$. Any power other than $3$ will grow much faster or much slower than the square pyramid numbers. $J=3$ will be off by the factor $3$ and the smaller terms