Solve the equation

a^{x} + b^{x} = 1

where

0 < a, b < 1

and

a + b < 1

, in the special cases:
(i)

a = b

(ii)

a = \frac{1}{2}, b = \frac{1}{4}

(iii)

a = \frac{1}{2}, b = \frac{1}{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.