Logarithmic graphsAQA A-Level Maths: Subtopic test

1. 1
   Variables $x$ and $y$ are related by $y=ax^n$, where $a$ and $n$ are constants. A graph of $\lg y$ against $\lg x$ is a straight line passing through the point $(0,0.6)$ on the vertical axis, with gradient $1.5$.

   (a)

   Which equation links $\lg y$ and $\lg x$?

   [1 mark]

   • A$y=1.5\lg x+0.6$
   • B$\lg y=0.6\lg x+1.5$
   • C$\lg y=1.5\lg x+10^{0.6}$
   • D$\lg y=1.5\lg x+0.6$

   (b)

   Find the value of $a$ to 3 significant figures.

   [1 mark]

   • A$0.6$
   • B$1.82$
   • C$3.98$
   • D$6$

   (c)

   Estimate the value of $y$ when $x=100$, giving your answer to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The number of cells $y$ in a culture $x$ days after the start is modelled by $y=kb^x$, where $k$ and $b$ are constants. A graph of $\lg y$ against $x$ is a straight line passing through $(0,2)$ and $(4,3.2)$.

   (a)

   What is the value of $\lg b$?

   [1 mark]

   • A$0.3$
   • B$1.2$
   • C$2$
   • D$0.8$

   (b)

   Find the value of $b$ to 3 significant figures.

   [1 mark]

   • A$0.3$
   • B$2.00$
   • C$1.35$
   • D$100$

   (c)

   Use the model to estimate how many days it takes for the culture to reach 5000 cells. Give your answer to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Variables $x$ and $y$ are thought to satisfy $y=ax^n$, where $a$ and $n$ are constants. A graph of $\lg y$ against $\lg x$ has a line of best fit passing through the points $(0.60,1.38)$ and $(1.20,2.28)$.

   (a)

   Show that $\lg y=n\lg x+\lg a$, and find the value of $n$.

   [3 marks]

   (b)

   Find the value of $a$, and hence estimate $y$ when $x=50$, giving your answer to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The number of monthly visitors $y$ to a website, $x$ months after it was launched, is modelled by $y=kb^x$, where $k$ and $b$ are constants. A graph of $\lg y$ against $x$ has a line of best fit passing through the points $(1,2.30)$ and $(5,3.10)$. These readings come from the first 5 months of data.

   (a)

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

   [6 marks]

   (b)

   (i) Estimate the number of visitors after 10 months.
   (ii) Estimate when the number of visitors first reaches 50 000.
   (iii) Comment on the reliability of using the model to predict the number of visitors after 40 months.

   [6 marks]

   Total for question 4: 12 marks

End of questions