file.md

The given data points are:
$x_0 = 654$, $y_0 = \log_{10} 654 = 2.8156$
$x_1 = 658$, $y_1 = \log_{10} 658 = 2.8182$
$x_2 = 659$, $y_2 = \log_{10} 659 = 2.8189$
$x_3 = 661$, $y_3 = \log_{10} 661 = 2.8202$

We use Newton's divided difference interpolation formula:
$P_3(x) = f[x_0] + f[x_0, x_1](x-x_0) + f[x_0, x_1, x_2](x-x_0)(x-x_1) + f[x_0, x_1, x_2, x_3](x-x_0)(x-x_1)(x-x_2)$

First divided differences:
$f[x_0, x_1] = \frac{y_1 - y_0}{x_1 - x_0} = \frac{2.8182 - 2.8156}{658 - 654} = \frac{0.0026}{4} = 0.00065$
$f[x_1, x_2] = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2.8189 - 2.8182}{659 - 658} = \frac{0.0007}{1} = 0.0007$
$f[x_2, x_3] = \frac{y_3 - y_2}{x_3 - x_2} = \frac{2.8202 - 2.8189}{661 - 659} = \frac{0.0013}{2} = 0.00065$

Second divided differences:
$f[x_0, x_1, x_2] = \frac{f[x_1, x_2] - f[x_0, x_1]}{x_2 - x_0} = \frac{0.0007 - 0.00065}{659 - 654} = \frac{0.00005}{5} = 0.00001$
$f[x_1, x_2, x_3] = \frac{f[x_2, x_3] - f[x_1, x_2]}{x_3 - x_1} = \frac{0.00065 - 0.0007}{661 - 658} = \frac{-0.00005}{3}$

Third divided difference:
$f[x_0, x_1, x_2, x_3] = \frac{f[x_1, x_2, x_3] - f[x_0, x_1, x_2]}{x_3 - x_0} = \frac{-0.00005/3 - 0.00001}{661 - 654} = \frac{-0.00005 - 0.00003}{3 \times 7} = \frac{-0.00008}{21}$

We want to find $P_3(656)$.
$x - x_0 = 656 - 654 = 2$
$x - x_1 = 656 - 658 = -2$
$x - x_2 = 656 - 659 = -3$

$P_3(656) = f[x_0] + f[x_0, x_1](x-x_0) + f[x_0, x_1, x_2](x-x_0)(x-x_1) + f[x_0, x_1, x_2, x_3](x-x_0)(x-x_1)(x-x_2)$
$P_3(656) = 2.8156 + (0.00065)(2) + (0.00001)(2)(-2) + (\frac{-0.00008}{21})(2)(-2)(-3)$
$P_3(656) = 2.8156 + 0.0013 + (0.00001)(-4) + (\frac{-0.00008}{21})(12)$
$P_3(656) = 2.8156 + 0.0013 - 0.00004 - \frac{12 \times 0.00008}{21}$
$P_3(656) = 2.8169 - 0.00004 - \frac{0.00096}{21}$
$P_3(656) = 2.81686 - \frac{0.00096}{21}$

Now we calculate the value:
$\frac{0.00096}{21} = \frac{96}{2100000} = \frac{3}{65625}$
$P_3(656) = 2.81686 - \frac{3}{65625} = \frac{281686}{100000} - \frac{3}{65625}$
$LCM(100000, 65625) = 2100000$
$P_3(656) = \frac{281686 \times 21}{2100000} - \frac{3 \times 32}{2100000} = \frac{5915406 - 96}{2100000} = \frac{5915310}{2100000} = \frac{591531}{210000} = \frac{197177}{70000}$

Converting the fraction to decimal:
$\frac{197177}{70000} \approx 2.8168142857...$

Rounding to 5 decimal places, we get 2.81681.

The final answer is $\boxed{2.81681}$.