Laws of logarithmsEdexcel International A Level Maths: Subtopic test

1. 1
   Given that $\log_a 2=p$ and $\log_a 5=q$, where $a$ is a positive constant and $a\neq1$.

   (a)

   Find $\log_a\left(\frac52\right)$ in terms of $p$ and $q$.

   [1 mark]

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

   (b)

   Find $\log_a20$ in terms of $p$ and $q$.

   [1 mark]

   • A$2p+q$
   • B$2pq$
   • C$4p+q$
   • D$p^2+q$

   (c)

   Express $\log_a\left(\frac{25}{8}\right)$ in terms of $p$ and $q$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   $x$ and $y$ are positive real numbers and $a$ is a positive constant with $a\neq1$. $\log_a x=3$ and $\log_a y=5$.

   (a)

   Find the value of $\log_a\left(\frac{x^2}{y}\right)$.

   [1 mark]

   • A$\frac65$
   • B$11$
   • C$-1$
   • D$1$

   (b)

   Find the value of $\log_a\left(\frac{a}{\sqrt y}\right)$.

   [1 mark]

   • A$-4$
   • B$-\frac32$
   • C$-\frac52$
   • D$\frac32$

   (c)

   Find the value of $\log_a\left(\frac1{x^2y}\right)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Throughout this question $a$ is a positive constant with $a\neq1$. Do not use a calculator.

   (a)

   Show that $\log_a48-\log_a3=4\log_a2$.

   [3 marks]

   (b)

   Write $3\log_a2+\log_a12-\frac12\log_a9$ as a single logarithm.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A student claims that, for all positive $x$ and $y$ and all $a>0$ with $a\neq1$: (I) $\log_a(x+y)=\log_a x+\log_a y$; (II) $\log_a(x^2)=\left(\log_a x\right)^2$; (III) $\log_a\left(\frac xy\right)=\frac{\log_a x}{\log_a y}$. The student also has to find the value of $\log_a\left(\frac{x^3}{y^2}\right)+\frac12\log_a\left(xy^4\right)-\log_a a$ when $\log_a x=2$ and $\log_a y=-3$.

   (a)

   Show, by a counter-example with $a=2$, that each of the statements (I), (II) and (III) is false.

   [6 marks]

   (b)

   Find the value of the logarithmic expression given at the end of the scenario.

   [6 marks]

   Total for question 4: 12 marks

End of questions