(i) If 2ax = 1 then log2 + x loga = 0 so
x = log2
log1/a
 .
(ii) To solve 1/2x + 1/4x = 1 write y = (1/2)x then y2 + y = 1 so
y = -1 ±Ö5
2

. It follows that
x log0.5 = log Ö5 - 1
2
giving x = 0.69424 to 5 significant figures.
Note that this method introduces an extraneous root of the quadratic equation because y = (1/2)x is always positive.

(iii) Solving 1/2x + 1/3x = 1, numerical approximation methods give x=0.7878849 to seven sig. figs.

Using the Newton Raphson method: f(x) = ax + bx - 1, f¢(x)=axln a + bxln b and
xn+1 = xn - f(xn)
f¢(xn)
 .