Logarithms and their lawsEdexcel A-Level Maths: Subtopic test

1. 1
   Consider the logarithms $\log_5125$ and $\log_2\frac18$.

   (a)

   Find the value of $\log_5125$.

   [1 mark]

   • A$3$
   • B$25$
   • C$\frac13$
   • D$5$

   (b)

   Find the value of $\log_2\frac18$.

   [1 mark]

   • A$3$
   • B$4$
   • C$-3$
   • D$-\frac13$

   (c)

   Find the value of $\log_48$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $p=\log_a2$ and $q=\log_a5$, where $a>1$.

   (a)

   Which expression is equal to $\log_a10$?

   [1 mark]

   • A$pq$
   • B$p+q$
   • C$\frac pq$
   • D$10p$

   (b)

   Which expression is equal to $\log_a0.4$?

   [1 mark]

   • A$q-p$
   • B$\frac pq$
   • C$p+q$
   • D$p-q$

   (c)

   Express $\log_a\sqrt{20}$ in terms of $p$ and $q$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The value $\pounds V$ of one savings account after $t$ years is $V=2000e^{0.04t}$. A second account has value $\pounds W=1500\times1.05^t$ after $t$ years.

   (a)

   Find how many years it takes for the first account to reach $\pounds3000$, giving your answer to 3 significant figures.

   [3 marks]

   (b)

   Show that the second account overtakes the first, and find after how many years, to 3 significant figures, the two accounts have equal value.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Positive numbers $x$ and $y$ satisfy $\log_3x+\log_3y=3$ and $\log_3x-\log_3y=1$.

   (a)

   Find the values of $x$ and $y$.

   [6 marks]

   (b)

   (i) Find the exact value of $\log_3(x^2y)$.
   (ii) Find the exact value of $\log_3\left(\dfrac{1}{\sqrt y}\right)$.
   (iii) Solve $\log_3(2z-1)=\log_3x+\log_3y-2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions