Solving equations of the form a^x = bEdexcel International A Level Maths: Subtopic test

1. 1
   The equation $5^x=40$ has solution $x=k$.

   (a)

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

   [1 mark]

   • A$\log_{10}(5x)=\log_{10}40$
   • B$x+\log_{10}5=\log_{10}40$
   • C$\left(\log_{10}5\right)^x=\log_{10}40$
   • D$x\log_{10}5=\log_{10}40$

   (b)

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

   [1 mark]

   • A$8.00$
   • B$0.436$
   • C$2.29$
   • D$1.60$

   (c)

   Solve $5^{x-2}=40$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the equation $3^{2x}=200$, where $x$ is a real number.

   (a)

   Which of the following is the exact solution of the equation?

   [1 mark]

   • A$x=2\log_3200$
   • B$x=\frac12\log_3200$
   • C$x=\log_3100$
   • D$x=\frac12\log_{200}3$

   (b)

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

   [1 mark]

   • A$2.41$
   • B$4.82$
   • C$33.3$
   • D$0.415$

   (c)

   Hence, or otherwise, solve $3^{2x+1}=600$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Give non-exact answers to 3 significant figures. A calculator may be used.

   (a)

   Solve $7^x=2^{x+3}$.

   [3 marks]

   (b)

   Show that the solution of $9^x=100$ can be written $x=\log_310$, and find its value to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Anna invests £2000 in an account paying 4% compound interest per year, so after $n$ years her account holds $V=2000\times1.04^n$ pounds. Ben invests £1500 in an account paying 6% compound interest per year, so after $n$ years his account holds $W=1500\times1.06^n$ pounds.

   (a)

   (i) Find the value of $n$, to 3 significant figures, for which Anna's account is worth exactly £3000.
   (ii) State the number of complete years after which Anna's account is first worth more than £3000.
   (iii) Find the value of her account after that number of years, to the nearest pound.

   [6 marks]

   (b)

   Ben claims that his account will be worth more than Anna's account within 15 years. Evaluate Ben's claim.

   [6 marks]

   Total for question 4: 12 marks

End of questions