Draw a logarithmic pattern and regress mass on time

For $y=a+b\ln x$ with $b<0$, $dy/dx=b/x<0$ and $d^2y/dx^2=-b/x^2>0$. Draw six positive points with equally spaced horizontal coordinates, decreasing steeply at first and then flattening. One illustrative choice is $x=1,2,3,4,5,6$ and $y=8-2\ln x$, all positive. These are authored example coordinates, not an extra printed table.

Plot all seven table pairs with time $t$ in seconds horizontally and mass $x$ in grams vertically. The comparison sketch below is for these measured data, distinct from the illustrative sketch in(a).

$r_{t,x}=0.9792220385\approx0.979$. There is a strong positive linear association between elapsed time and mass in this sample.

Multiplying every mass by the positive factor 1000 leaves $r$ unchanged. Its value does not depend on a positive change of measurement scale.

Time is the controlled explanatory variable, and mass is the response to estimate. Regress $x$ on $t$: $x=4.248360656+0.2056557377t$, or $x=0.206t+4.25$ (3 s.f.).

At $t=10$, the unrounded line gives $\hat x=4.248360656+0.2056557377(10)=6.304918033$ g, which is $6.30$ g (3 s.f.). The printed answer is $6.31$ g; rounding coefficients first gives $6.305$ from the printed 5-s.f. line, or $6.31$ from the 3-s.f. line $0.206t+4.25$. Retaining full precision avoids this double-rounding difference. Both source result and rounding distinction are recorded. Since 10 lies within $[5,40]$ and $r$ is close to 1, this is a reasonable interpolated estimate under the linear model.