Calculating the binary value of log10 of x

First, divide $x$ by $10^n$, where $n$ is chosen such that $1\le x/10^n<10$. The integer part of ${\log }_{10}(x)$ is given by $n$. The binary expansion of the fractional part can be determined by the following procedure.

Start by setting $x_0=x/10^n$. To determine $b_k$ (the $k$th digit of the binary expansion):

• $b_k=0$ if $x_{k-1}^2<10$. In this case, $x_k=x_{k-1}^2$.
• $b_k=1$ if $x_{k-1}^2\ge 10$. In this case, $x_k=x_{k-1}^2/10$