1.42: Log graph and graphical roots

(a) Complete the table of values for $y=\ln(3x+1)+2$, giving your answers to 2 decimal places.

(b) On the grid opposite, draw the graph of $y=\ln(3x+1)+2$ for $0\leqslant x\leqslant 6$.

(c) Use your graph to obtain an estimate, to 1 decimal place, for the value of $\ln 10.6$.

You must show clearly how you have used your graph.

(d) By drawing a straight line on the grid, obtain estimates, to 1 decimal place, for the roots of the equation $(3x+1)^{2}=\mathrm{e}^{(x+1)}$ in the interval $0\leqslant x\leqslant 6$.