$\log_{10}x \approx x^{ \frac{1}{2^{19}}}227695 - 227695$
I plotted the graphs of both $\log_{10}x$ and $x^{ \frac{1}{2^{19}}} 227695 - 227695$
And found that they almost overlapped.
Can you give a explanation of this or can you find how this approximation is approached.
The approximation is of the form $\log_{10}x \approx a(x^b-1)$ for $b \ll 1$. This form assures that the value at $x=1$ is $0$, which agrees with the logarithm. The derivative is $abx^{b-1}$ while the derivative of $\log_{10}x$ is $\frac 1{x\log 10}$ When $b \ll 1$ and $x$ is not too large $x^b$ will be very close to $1$. We note that $$227695 \cdot 2^{-19}-\frac 1{\log 10} \approx 10^{-7}$$ so the derivatives are very close, which will make the curves very close until the $x^{2^{-19}}$ factor starts to matter. The approximation will be very good for any tiny $b$ when $a$ is chosen to make $ab \approx \frac 1{\log 10}$ The smaller $b$ gets the longer it will stay close. Once $x$ gets large enough the approximation will be terrible.