Assume that $k<1$ and an iteration satisfies $|a_{n+1}− t| < k|a_n − t| \Rightarrow |a_{n}− t| < k^n|a_0 − t|$
And here the book declares that iteration step contributes at worst roughly the same number of additional correct decimal digits, the number obtained in each step being approximately $− \log_{10} k$.
I am trying to understand relation between correct decimal digits and logarithm. We have not defined rigorously logarithm function but I remember basic properties of logarithm from school math. So I could write $\log_{10}|a_n − t|<n \log_{10}k+\log_{10}|a_0 − t|$
But how this inequality implies to the decimal digits I am troubling imagine. Could anyone explain me, please. Thank you in advance.
An example can be ilustrative:
Take $x = .12345678$ and $y = .12344321$
Since they have in common the first $\color{red}4$ digits, we have $|x-y| = .00001357 < .0001 =10^{-4}$, so $\log_{10}|x-y| < -\color{red}{4}$
Can you see the relation between correct number of digits of an approximation and $\log_{10}$ now?