Solve the equation exponential radical Solve the equation$$31+8\sqrt{15}=(4+\sqrt{15})^x$$for $x$.
I think you could set up a recursion from the coefficients of $(4+\sqrt{15})^{x}$ to the coefficients of $(4+\sqrt{15})^{x+1}$ then find the general formula using the characteristic equation?
It should look like $a_{n+1}=4a_{n}+15b_{n}$ and $b_{n+1}=4b_{n}+a_{n}$, cancel out one of variables, and then take the characteristic equation.

Using your suggested approach:
$$(4+\sqrt{15})^n=a_n+b_n\sqrt{15}$$
$$a_0=1,b_0=0$$
\begin{align}a_{n+1}+b_{n+1}\sqrt{15}&=(4+\sqrt{15})^{n+1}\\&=(4+\sqrt{15})(a_n+b_n\sqrt{15})\\&=(4a_n+15b_n)+(a_n+4b_n)\sqrt{15}\end{align}
It then suffices to work out a few values:
\begin{array}{c|c|c}n&a_n&b_n\\\hline0&1&0\\1&4&1\\\color{#4477FF}2&\color{#4477FF}{31}&\color{#4477FF}8\end{array}

As an alternative, one could instead numerically approximate using $\sqrt{15}\approx\sqrt{16}=4$ to see that we roughly want to solve
$$31+8\cdot4\approx8^n$$
from which the answer is clear, assuming it must be an integer.
