The tangent to the curve $y=3\ln(x-1)$ at $A$, where $x=4$, meets the $x$-axis at $B$

Question

The tangent to the curve $y=3\ln(x-1)$ at $A$, where $x=4$, meets the $x$-axis at $B$. The curve meets the $x$-axis at $C$. Find the equation of the tangent at $A$.[3] Find the coordinates of $B$ and $C$.[3] Hence find the exact area enclosed by the curve, the tangent and the $x$-axis.[5]

Accepted Answer

(i) $y=x-4+3\ln3$ (ii) $B=(4-3\ln3,0)$, $C=(2,0)$ (iii) $\dfrac92(\ln3)^2-9\ln3+6\text{ units}^2$

Logarithmic curve with a tangent and the x-axis

Logarithmic curve with a tangent and the x-axis

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