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Log Attack

Age 16 to 18 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).
More generally you will need to use a numerical method for finding approximate solutions. See Equation Attack.