sum powers with constant exponent I'd like to know if there is a solution to the following equation and how to solve it, thank you.
$$a^t + b^t + \cdots + z^t = Q$$
I want to find $t$, knowing $a,b,\cdots,z$ and $Q$
(Also is there a denomination to that sum)
Best Regards, FG
 A: I do not think that you could get an analytical solution but numerically the problem is solvable quite efficiently. Considering that you look for the zero of
$$f(t)=\sum_{i=1}^n a_i^t- Q$$
it is better to consider instead the function
$$g(x)=\log\left(\sum_{i=1}^{n} a_i^t\right)-\log(Q)$$ which is much better conditioned (it is almost a straight line).
An estimate of the zero could be obtained using a Taylor expansion of the logarithm  at $t=0$ giving
$$t_0=\frac {n(\log(Q) -\log(n)) } {\sum_{i=1}^{n} \log(a_i)}$$ but, since is equivalent to the first iteration of Newton method, we can simply use $t_0=0$.
Let us try with $n=12$, $a_i=p_i$ and $Q=123456789$. Using Newton method, the iterates would be
$$\left(
\begin{array}{cc}
 n & t_n \\
 0 & 0 \\
 1 & 6.538079375 \\
 2 & 4.994362680 \\
 3 & 4.986570543 \\
 4 & 4.986570264
\end{array}
\right)$$
which the solution for ten significant figures.
If we repeat the problem using now $a_i=(-1)^i p_i$, the problem is more delicate starting at $t=0$ because of some indeterminations. Starting with $t=\frac 1 {10}$ would give as Newton iterates
$$\left(
\begin{array}{cc}
 n & t_n \\
 0 & 0.100000000 \\
 1 & 1.762045905 \\
 2 & 5.138386859 \\
 3 & 5.216134407 \\
 4 & 5.216145136 \\
 5 & 5.216145136
\end{array}
\right)$$
Edit
Since in the question, instead of using $a_i$ as notation, you used $(a,b,c,\cdots,x,y,z)$, let us do it for $n=26$, $a_i=p_i$ and $Q=12345^{6789}$. The iterates will be
$$\left(
\begin{array}{cc}
 n & t_n \\
 0 & 0 \\
 1 & 18822.60277 \\
 2 & 13858.62238
\end{array}
\right)$$
which the solution for ten significant figures.
