The natural logarithm ln xEdexcel International A Level Maths: Subtopic test

1. 1
   The function $f$ is defined by $f(x)=\ln(3x-2)$, where $x$ takes the largest possible domain.

   (a)

   What is the largest possible domain of $f$?

   [1 mark]

   • A$x>\frac23$
   • B$x>0$
   • C$x>-\frac23$
   • D$x\ge\frac23$

   (b)

   What is the solution of $f(x)=1$?

   [1 mark]

   • A$x=\frac{e-2}{3}$
   • B$x=\frac{e}{3}-2$
   • C$x=1$
   • D$x=\frac{e+2}{3}$

   (c)

   Find the exact solution of $f(x)=-2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The functions $g$ and $h$ are defined by $g(x)=e^{x}$ and $h(x)=\ln x$, for $x>0$ in the case of $h$.

   (a)

   The graph of $y=h(x)$ is a reflection of the graph of $y=g(x)$ in which line?

   [1 mark]

   • Athe $x$-axis
   • B$y=x$
   • Cthe $y$-axis
   • D$y=-x$

   (b)

   What is the value of $h(g(5))+g(h(5))$?

   [1 mark]

   • A$5$
   • B$25$
   • C$10$
   • D$e^{5}+\ln5$

   (c)

   State the equation of the asymptote of the graph of $y=h(x)$ and the coordinates of the point where it crosses the $x$-axis.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The curve $C$ has equation $y=\ln(2x+5)$.

   (a)

   Show that the solution of $\ln(2x+5)=3$ is $x=\frac{e^{3}-5}{2}$ and find its value to 3 significant figures.

   [3 marks]

   (b)

   The curve $C$ meets the $y$-axis at $P$ and the $x$-axis at $Q$. Find the exact coordinates of $P$ and $Q$, and state the equation of the vertical asymptote of $C$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The height $H$ metres of a young tree, $t$ years after it was planted, is modelled by $H=2\ln(3t+1)+0.5$ for $t\ge0$.

   (a)

   (i) State the height of the tree when it is planted.
   (ii) Find the height of the tree $5$ years after it was planted, to 3 significant figures.
   (iii) Find the time taken for the tree to reach a height of $8$ m, to 3 significant figures.

   [6 marks]

   (b)

   A botanist states that trees of this species reach a height of $15$ m within $30$ years of planting. Evaluate whether the model supports this statement, and comment on the suitability of the model for predicting the height of the tree in the long term.

   [6 marks]

   Total for question 4: 12 marks

End of questions