Your attachment is hard to read but it says:
"Find real root of by iteration method
The answer is
The answer is 3.7893."
How did he get ? Pretty much an "educated" guess. Under the right conditions, if 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 ." Then I would get:
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 , my first step would have been
And then the iteration would be the same as above.
If I had started with something really bad, say , then I would have
and I am heading right back to 3.7893.