Logarithms and their lawsAQA A-Level Maths: Subtopic test

1. 1
   In this question, evaluate each logarithm without using a calculator.

   (a)

   Find the value of $\log_381$.

   [1 mark]

   • A$3$
   • B$4$
   • C$27$
   • D$\frac14$

   (b)

   Find the value of $\log_2\frac18$.

   [1 mark]

   • A$3$
   • B$-\frac13$
   • C$-3$
   • D$\frac18$

   (c)

   Find the exact value of $\log_48$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Let $a=\log_{10}2$ and $b=\log_{10}3$.

   (a)

   Write $\log_{10}12$ in terms of $a$ and $b$.

   [1 mark]

   • A$a+b$
   • B$2ab$
   • C$a+2b$
   • D$2a+b$

   (b)

   Write $\log_{10}\frac16$ in terms of $a$ and $b$.

   [1 mark]

   • A$-a-b$
   • B$\frac{1}{a+b}$
   • C$b-a$
   • D$a-b$

   (c)

   Write $\log_{10}\sqrt{18}$ in terms of $a$ and $b$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The positive real numbers $x$ and $y$ are such that $\ln x=p$ and $\ln y=q$.

   (a)

   Express $\ln\left(\dfrac{x^3}{\sqrt y}\right)$ in terms of $p$ and $q$.

   [3 marks]

   (b)

   Given also that $\ln\left(\dfrac{x^3}{\sqrt y}\right)=5$ and $\ln(xy)=4$, find $p$ and $q$, and hence find the exact values of $x$ and $y$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In this question, $x$ is real and the argument of every logarithm must be positive.

   (a)

   Solve $\log_2(x+6)-\log_2(x-1)=3$.

   [6 marks]

   (b)

   Solve $2\log_3x-\log_3(x+4)=1$, giving your answer in exact form.

   [6 marks]

   Total for question 4: 12 marks

End of questions