Logarithmic graphs and modellingEdexcel International A Level Maths: Subtopic test

1. 1
   A scientist believes that two variables are related by $y=ax^{n}$, where $a$ and $n$ are constants. She plots $\lg y$ against $\lg x$ and obtains a straight line with gradient $3$ that crosses the vertical axis at $2$.

   (a)

   Which equation is obtained by taking logarithms to base $10$ of $y=ax^{n}$?

   [1 mark]

   • A$\lg y=a+n\lg x$
   • B$\lg y=n\lg a\times\lg x$
   • C$\lg y=\lg a+n\lg x$
   • D$\lg y=\lg a+x\lg n$

   (b)

   What is the value of $a$?

   [1 mark]

   • A$2$
   • B$100$
   • C$20$
   • D$0.01$

   (c)

   Find the value of $y$ when $x=4$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two variables are related by $y=kb^{x}$, where $k$ and $b$ are positive constants. A graph of $\lg y$ against $x$ is a straight line through the points $(0,\,0.6)$ and $(5,\,2.1)$.

   (a)

   What does the gradient of the straight line represent?

   [1 mark]

   • A$\lg b$
   • B$b$
   • C$\lg k$
   • D$k$

   (b)

   What is the value of $k$ to 3 significant figures?

   [1 mark]

   • A$0.6$
   • B$0.25$
   • C$1.82$
   • D$3.98$

   (c)

   Use the model to find the value of $y$ when $x=8$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The variables $x$ and $y$ are related by $y=ax^{n}$. A graph of $\lg y$ against $\lg x$ is a straight line through the points $(0.2,\,0.8)$ and $(0.6,\,1.6)$.

   (a)

   Show that $n=2$ and find the value of $a$ to 3 significant figures.

   [3 marks]

   (b)

   Find the value of $x$ for which $y=500$. Give your answer to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The number $P$ of bacteria in a culture, $t$ hours after the start of an experiment, is modelled by $P=kb^{t}$. A graph of $\lg P$ against $t$ is a straight line through the points $(2,\,3.4)$ and $(6,\,4.2)$.

   (a)

   (i) Show that $\lg P=\lg k+t\lg b$.
   (ii) Find the values of $k$ and $b$, giving $b$ to 3 significant figures.

   [6 marks]

   (b)

   (i) Find the number of bacteria predicted after $10$ hours.
   (ii) Find the time at which the model predicts $10^{6}$ bacteria.
   (iii) Comment on the validity of the model for large values of $t$.

   [6 marks]

   Total for question 4: 12 marks

End of questions