The equation 8^x + 13^x = 108 cannot be solved algebraically, as it does not yield to logarithmic methods or other traditional algebraic techniques. The only way to find a solution is through numerical approximation, with the approximate value of x being 1.68279. The Newton-Raphson method is suggested as a viable numerical approach for finding this approximation. Overall, it is emphasized that most transcendental equations, like this one, lack exact solutions. Therefore, numerical methods are the only practical means to solve such equations.
