Question

Solve by iteration method:
(i) 1+logx=x

Answer 1

$$x=g(x)=1+\log x$$

$x_0=0.5$

$$x_{i+1}=g(x_i)=1+\log x_i, i=0, 1,2,...$$

$$\def\arraystretch{1.5} \begin{array}{c:c:c} i & x_i & g(x_i) \\ \hline 0 & 0.5 & 0.69897000 \\ 1 & 0.69897000 & 0.84445854 \\ 2 &0.84445854 & 0.92657833 \\ 3 & 0.92657833 & 0.96688214\\ 4 & 0.96688214 & 0.98537354 \\ 5 & 0.98537354 & 0.99360090\\ 6 & 0.99360090 & 0.99721198\\ 7 & 0.99721198 & 0.99878749\\ 8 & 0.99878749 & 0.99947309\\ 9 & 0.99947309 & 0.99977111\\ 10 & 0.99977111 & 0.99990058\\ 11 & 0.99990058 & 0.99995682\\ 12 & 0.99995682 & 0.99998125\\ 13 & 0.99998125 & 0.99999186\\ 14 & 0.99999186 & 0.99999646\\ 15 & 0.99999646 & 0.99999846\\ 16 & 0.99999846 & 0.99999933\\ 17 & 0.99999933 & 0.99999971\\ 18 & 0.99999971 & 0.99999987\\ 19 & 0.99999987 & 0.99999994\\ 20 &0.99999994 & 0.99999997\\ 21 & 0.99999997 & 0.99999999\\ 22 &0.99999999 & 1.00000000\\ \hdashline 23 &1.00000000 & \\ \end{array}$$

$x\approx1.00000000$