LogarithmsIB MYP Maths Extended: Subtopic test

1. 1
   Ana is converting between index form and logarithm form.

   (a)

   Which statement is equivalent to $3^4=81$?

   [1 mark]

   • A$\log_{4}81=3$
   • B$\log_{81}3=4$
   • C$\log_{3}81=4$
   • D$\log_{81}4=3$

   (b)

   Find the value of $\log_{2}32$.

   [1 mark]

   • A$5$
   • B$16$
   • C$64$
   • D$30$

   (c)

   Find the value of $\log_{5}\frac{1}{25}$.

   [2 marks]

   Total for question 1: 4 marks

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

   (a)

   Find $\log_{10}6$ in terms of $a$ and $b$.

   [1 mark]

   • A$ab$
   • B$a-b$
   • C$\frac ab$
   • D$a+b$

   (b)

   Find $\log_{10}8$ in terms of $a$ and $b$.

   [1 mark]

   • A$a+3$
   • B$3a$
   • C$a^3$
   • D$8a$

   (c)

   Write $\log_{10}\frac43$ in terms of $a$ and $b$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A bacterial culture starts with $500$ cells. In dish A the number of cells doubles every hour, so after $t$ hours there are $N=500\times2^{t}$ cells. A calculator may be used.

   (a)

   Find when the culture first reaches $3500$ cells. Give $t$ correct to 3 significant figures.

   [3 marks]

   (b)

   In dish B the culture starts with $500$ cells and triples every hour, so $N=500\times3^{t}$. Find the time for the culture in dish B to reach $20\,000$ cells, correct to 3 significant figures, and verify your answer by substitution.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The sound level $L$ (in decibels, dB) of a sound of intensity $I$ (in watts per square metre) is modelled by $L=10\log_{10}\left(\frac{I}{10^{-12}}\right)$. A calculator may be used.

   (a)

   (i) Find the sound level when $I=10^{-6}$.
   (ii) Show that when the intensity doubles, the sound level increases by $10\log_{10}2$, and find this increase correct to 3 significant figures.

   [6 marks]

   (b)

   A concert has a sound level of $110$ dB. Hearing damage can occur after long exposure to $85$ dB. (i) Find the intensity at $110$ dB. (ii) Find the intensity at $85$ dB. (iii) Use your answers to justify why ear protection is recommended at the concert.

   [6 marks]

   Total for question 4: 12 marks

End of questions