1.5 Laws of exponents and introduction to logarithmsIB Maths: Applications and Interpretation SL: Subtopic test

1. 1
   In this question, $x\neq0$.

   (a)

   Simplify $x^{5}\times x^{-8}$.

   [1 mark]

   • A$x^{-3}$
   • B$x^{13}$
   • C$x^{-13}$
   • D$x^{3}$

   (b)

   Which expression is equal to $(3x)^{-2}$?

   [1 mark]

   • A$\frac{1}{3x^{2}}$
   • B$-9x^{2}$
   • C$\frac{1}{9x^{2}}$
   • D$\frac{x^{2}}{9}$

   (c)

   Write $\frac{(2x^{3})^{4}}{8x^{5}}$ in the form $kx^{n}$, where $k$ and $n$ are integers.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Logarithms with base $10$ and base $e$ are the inverse functions of $10^{x}$ and $e^{x}$.

   (a)

   Find the value of $\log_{10}0.01$.

   [1 mark]

   • A$2$
   • B$-2$
   • C$-100$
   • D$-10$

   (b)

   Find the value of $\ln\left(e^{3}\right)+\log_{10}10$.

   [1 mark]

   • A$3$
   • B$13$
   • C$e^{3}+10$
   • D$4$

   (c)

   Use your GDC to write down the value of $\log_{10}250$ and the value of $\ln250$, each correct to 3 significant figures.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A radioactive substance decays so that its mass, $m$ grams, after $t$ years is modelled by $m=80e^{-0.05t}$.

   (a)

   Write down the initial mass, and use your GDC to find the mass after 10 years.

   [3 marks]

   (b)

   Find the time taken for the mass to fall to 20 g. Give your answer correct to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The pH of a solution is defined as $\mathrm{pH}=-\log_{10}h$, where $h$ is the concentration of hydrogen ions in $\mathrm{mol\,dm^{-3}}$.

   (a)

   (i) Calculate the pH of a solution with $h=3.2\times10^{-4}$.
   (ii) A solution has pH $5.6$. Find the value of $h$.
   (iii) Show that when the pH of a solution decreases by $1$, the value of $h$ is multiplied by $10$.

   [6 marks]

   (b)

   Solution $S$ has pH $2.8$. Solution $T$ has a hydrogen ion concentration $25$ times that of $S$.
   (i) Find $h$ for solution $S$.
   (ii) Find $h$ for solution $T$.
   (iii) Find the pH of solution $T$, and state which solution is more acidic.

   [6 marks]

   Total for question 4: 12 marks

End of questions