= Solution
Let $s\geq0$ be distance along the unit direction from part (a). The point is $\mathbf r(s)=(1,2)+s(1,2)/\sqrt5$, whose distance from the origin is $\sqrt5+s$. For <finite increments of a logarithmic radial function>, the exact change is
$$
\Delta T=\ln\frac{(\sqrt5+s)^2}{5}=2\ln\left(1+\frac s{\sqrt5}\right).
$$
Thus an increase of $0.2$ requires
$$
\boxed{s=\sqrt5(e^{0.1}-1).}
$$
If the intended calculation is the local <directional derivative> estimate, its first-order value is $s\simeq\Delta T/|\nabla T|=\boxed{\sqrt5/10}$. The latter approximates the exact finite distance rather than equaling it. Numerically they are about $0.235$ and $0.224$, respectively.
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