Estimate the error of interpolating ,,, Estimate the error of interpolating (${lnx}$) . at ${x=3}$ with an interpolation polynomial with base points ${x=1 , x=2 , x=4 , x=6 }$ .

Well, the polynomial $p$ that you need has
$$
p(1) = ln(1) = 0\\
p(2) = ln(2)\\
p(4) = ln(4) = 2 ln(2) \\
p(6) = ln(6)
$$
Writing 
$$
p(x) = 
a_1 \frac{(x-2)(x-4)(x-6)}{(x-1)(x-2)(x-4)(x-6)}+
a_2 \frac{(x-1)(x-4)(x-6)}{(x-1)(x-2)(x-4)(x-6)}+
a_3 \frac{(x-1)(x-2)(x-6)}{(x-1)(x-2)(x-4)(x-6)}+
a_4 \frac{(x-1)(x-2)(x-4)}{(x-1)(x-2)(x-4)(x-6)},
$$
you can see, by plugging in $x = 1, 2, 4,$ and $6$ that $a_1 = 0; a_2 = ln(2); a_3 = ln(4)$, and $a_4 = ln(6)$. So your polynomial is 
$$
p(x) = 
ln(2) \frac{(x-1)(x-4)(x-6)}{(x-1)(x-2)(x-4)(x-6)}+
ln(4) \frac{(x-1)(x-2)(x-6)}{(x-1)(x-2)(x-4)(x-6)}+
ln(6) \frac{(x-1)(x-2)(x-4)}{(x-1)(x-2)(x-4)(x-6)}.
$$
Plug in $x = 3$, and take the difference from $ln(3)$ and you've got your answer. 
