Answer: \(\displaystyle y=\sqrt[3]{2\ln x+3}\).

Answer: \(\displaystyle y=\sqrt[3]{2\ln x+3}\).

Since \(y^3\) is plotted against \(\ln x\), the straight line has equation

\(y^3=m\ln x+c.\)

The line passes through \((1,5)\) and \((6,15)\), so its gradient is

\(m=\frac{15-5}{6-1}=2.\)

Using the point \((1,5)\),

\(5=2(1)+c.\)

Thus

\(c=3.\)

Therefore

\(y^3=2\ln x+3.\)

Hence

\(y=\sqrt[3]{2\ln x+3}.\)