hello i have problem with solution. please clear my doubt.
my doubt is how did the writer get x0 = 3.8
how did he get x0 value.
Hi avengerevenge,
Maybe from graphing
$\displaystyle f(x)=2x-log_{10}x-7=0$
Or..
$\displaystyle 10^{2x-log_{10}x}=10^7$
$\displaystyle \frac{10^{2x}}{10^{log_{10}x}}>10^7$
$\displaystyle \frac{10^{2x}}{x}>10^7$
$\displaystyle 10^{2x}>x10^7$
$\displaystyle x=1,\ 10^2<10^7$
$\displaystyle x=2,\ 10^4<(2)10^7$
$\displaystyle x=3,\ 10^6<(3)10^7$
$\displaystyle x=4,\ 10^8>(4)10^7$
$\displaystyle 3<x<4$
Your attachment is hard to read but it says:
"Find real root of $\displaystyle 2x- log_{10}(x)= 7$ by iteration method
The answer is $\displaystyle x= \frac{1}{2}(log_{10}(x)+ 7)$
$\displaystyle x_0= 3.8$
$\displaystyle x_1= \frac{1}{2}(log_{10}(x)+ 7)= 3.7893$
$\displaystyle x_2= \frac{1}{2}(log_{10}(x)+ 7)= 3.7893$
The answer is 3.7893."
How did he get $\displaystyle x_0= 3.8$? Pretty much an "educated" guess. Under the right conditions, if $\displaystyle x_0$ is any number reasonably close to the answer to begin with, this sequence will converge to the solution to the equation. I suspect that this author already knew the answer and chose 3.8 because it was already correct to the first decimal place.
If I were coming to this equation without knowing the answer I might argue "log(x) is always smaller than x so what happens if I ignore it? The equation becomes 2x= 7 and x= 7/2= 3.5. I think I will try $\displaystyle x_0= 3.5$." Then I would get:
$\displaystyle x_0= 3.5$
$\displaystyle x_1= \frac{1}{2}(log_{10}(3.5)+ 7)= 3.7720$
$\displaystyle x_2= \frac{1}{2}(log_{10}(3.7720)+ 7)= 3.7883$
$\displaystyle x_3= \frac{1}{2}(log_{10}(3.7883)+ 7)= 3.7892$
$\displaystyle x_4= \frac{1}{2}(log_{10}(3.7892)+ 7)= 3.7893$
$\displaystyle x_5= \frac{1}{2}(log_{10}(3.7893)+ 7)= 3.7893$
Since those last two iterations are the same to the four decimal places I am keeping, I can stop here, confident that my answer is correct at least to three decimal places. It took a couple more iterations because my first value was not as close to the correct answer as 3.8 but I still get the same answer.
Notice that if I had started with $\displaystyle x_0= 1$, my first step would have been
$\displaystyle x_1= \frac{1}{2}(log_{10}(1)+ 7)= 7/2= 3.5$
And then the iteration would be the same as above.
If I had started with something really bad, say $\displaystyle x_0= 100$, then I would have
$\displaystyle x_1= \frac{1}{2}(log_{10}(100)+ 7)= 27/2= 13.5$
$\displaystyle x_2= \frac{1}{2}(log_{10}(13.5)+ 7)= 4.0652$
$\displaystyle x_3= \frac{1}{2}(log_{10}(4.0652)+ 7)= 3.8045$
$\displaystyle x_4= \frac{1}{2}(log_{10}(3.8045)+ 7)= 3.7901$
and I am heading right back to 3.7893.