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Equation 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 $ (iii) $a = {1\over 2}, \ b={1\over 3}$ .

Although you can find exact solutions in special cases like (i) and (ii) it soon becomes apparent that generally you will need to use a numerical method for finding approximate solutions.