Solving exponential equationsAQA A-Level Maths: Subtopic test

1. 1
   The equation $5^x=40$ is to be solved using logarithms.

   (a)

   Which expression gives the solution of $5^x=40$?

   [1 mark]

   • A$x=\lg8$
   • B$x=\frac{\lg5}{\lg40}$
   • C$x=\frac{\lg40}{\lg5}$
   • D$x=\lg40\times\lg5$

   (b)

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

   [1 mark]

   • A$2.29$
   • B$1.60$
   • C$0.436$
   • D$8$

   (c)

   Hence, or otherwise, solve $5^{x-1}=40$, giving your answer to 3 significant figures.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   £2000 is invested in an account that pays 3.5% compound interest per year. After $n$ complete years the value of the investment is £$2000\times1.035^n$.

   (a)

   Which equation must be solved to find when the investment first reaches £3000?

   [1 mark]

   • A$1.035n=1.5$
   • B$1.035^n=1.5$
   • C$2000+1.035^n=3000$
   • D$2000\times1.035^n=1500$

   (b)

   What is the smallest whole number of years after which the investment is worth more than £3000?

   [1 mark]

   • A$11$
   • B$15$
   • C$43$
   • D$12$

   (c)

   Find the smallest whole number of years after which the investment is worth more than £5000.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two equations are given: $4^{3x-1}=100$ and $3\mathrm{e}^{2x}=21$.

   (a)

   Solve $4^{3x-1}=100$, giving your answer to 3 significant figures.

   [3 marks]

   (b)

   Solve $3\mathrm{e}^{2x}=21$. Give the exact solution and its value to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two bacterial cultures are studied. In culture A the number of bacteria, $N_A$, after $t$ hours is modelled by $N_A=800\times1.6^t$. In culture B an antibiotic is added, and the number of bacteria, $N_B$, after $t$ hours is modelled by $N_B=5000\times0.85^t$.

   (a)

   (i) Write down the number of bacteria in culture A at $t=0$.
   (ii) Find the time at which culture A reaches 20 000 bacteria.
   (iii) Find the time taken for culture A to double in size.
   Give your times in hours to 3 significant figures.

   [6 marks]

   (b)

   (i) Find the time at which culture B falls to 500 bacteria.
   (ii) Find the time taken for culture B to halve in size.
   (iii) State, with a reason, whether this model ever predicts that culture B has no bacteria.
   Give your times in hours to 3 significant figures.

   [6 marks]

   Total for question 4: 12 marks

End of questions