Challenge Level

Solve the equation $a^x + b^x = 1$ where $0< a, b < 1$ and $a
+ b < 1$, in the special cases:

(i) $a = b\quad $ (ii) $a = {1\over 2}, \ b={1\over 4}\quad $

You can find exact solutions to the equation $a^x + b^x = 1$ in special cases like (i) and (ii).

(i) $a = b\quad $ (ii) $a = {1\over 2}, \ b={1\over 4}\quad $

You can find exact solutions to the equation $a^x + b^x = 1$ in special cases like (i) and (ii).

More generally you will need to use a numerical method for
finding approximate solutions. See
Equation Attack.