Exponential and logarithmic equationsEdexcel A-Level Maths: Subtopic test

1. 1
   Consider the equation $5^{x}=40$.

   (a)

   Find the value of $x$, to 3 significant figures.

   [1 mark]

   • A$8$
   • B$2.29$
   • C$2.08$
   • D$0.436$

   (b)

   Which expression gives the exact solution for $x$?

   [1 mark]

   • A$\frac{\ln5}{\ln40}$
   • B$\ln40-\ln5$
   • C$\frac{40}{5}$
   • D$\log_{5}40$

   (c)

   Show that $x=\frac{\log_{10}40}{\log_{10}5}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the equation $2^{3x-1}=3$.

   (a)

   Taking logarithms to base 10 of both sides, which equation is correct?

   [1 mark]

   • A$(3x-1)\log_{10}2=\log_{10}3$
   • B$3x-1=\frac{\log_{10}3}{2}$
   • C$(3x-1)\log_{10}2=3$
   • D$3x\log_{10}2-1=\log_{10}3$

   (b)

   Find the value of $x$, to 3 significant figures.

   [1 mark]

   • A$0.528$
   • B$0.195$
   • C$0.862$
   • D$2.58$

   (c)

   Solve $3^{2x+1}=50$, giving your answer to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The value, £$V$, of an investment after $t$ years is modelled by $V=2500\times1.04^{t}$.

   (a)

   Find the time taken for the value to reach £4000. Give your answer in years to 3 significant figures.

   [3 marks]

   (b)

   (i) Show that the value of the investment has doubled when $t=\frac{\ln2}{\ln1.04}$.
   (ii) Find the least whole number of years after which the value first exceeds £10 000.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two bacterial cultures are grown in a laboratory. After $t$ hours, culture A contains $N_A=500\times3^{t}$ cells and culture B contains $N_B=4000\times2^{t}$ cells.

   (a)

   (i) Show that the two cultures contain the same number of cells when $1.5^{t}=8$.
   (ii) Hence find the value of $t$, to 3 significant figures.
   (iii) Find the number of cells in each culture at this time, to 2 significant figures.

   [6 marks]

   (b)

   (i) Find the time at which culture A contains ten times as many cells as culture B. Give your answer to 3 significant figures.
   (ii) A student claims that after 12 hours culture A will contain more than 100 times as many cells as culture B. Evaluate this claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions