1.9 Laws of logarithmsIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   Let $a=\ln3$ and $b=\ln5$.

   (a)

   Which expression is equal to $\ln15$?

   [1 mark]

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

   (b)

   Which expression is equal to $\ln\frac{9}{5}$?

   [1 mark]

   • A$\frac{2a}{b}$
   • B$2a+b$
   • C$a^2-b$
   • D$2a-b$

   (c)

   Express $\ln\sqrt{45}$ in terms of $a$ and $b$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The sound level $L$ decibels (dB) of a noise of intensity $I$ W m$^{-2}$ is given by $L=10\log_{10}\left(\frac{I}{10^{-12}}\right)$.

   (a)

   Find the sound level of a noise with intensity $10^{-6}$ W m$^{-2}$.

   [1 mark]

   • A$60$ dB
   • B$6$ dB
   • C$120$ dB
   • D$-60$ dB

   (b)

   The intensity of a noise is multiplied by $100$. By how much does the sound level increase?

   [1 mark]

   • A$2$ dB
   • B$10$ dB
   • C$20$ dB
   • D$100$ dB

   (c)

   Use your GDC to find the intensity of a noise with sound level $85$ dB.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A culture of bacteria grows so that after $t$ hours there are $N=800e^{0.35t}$ bacteria.

   (a)

   Find the time taken for the population to double. Give your answer in hours to 3 significant figures.

   [3 marks]

   (b)

   Show that the time $T$ at which the population reaches $1\,000\,000$ satisfies $e^{0.35T}=1250$, and hence find $T$ in hours to 3 significant figures.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Biologists model the metabolic rate $R$ watts of mammals of mass $m$ kg by $R=km^{b}$, where $k$ and $b$ are constants. A GDC may be used.

   (a)

   (i) Show that $\ln R=\ln k+b\ln m$.
   (ii) For one group of mammals, a line of best fit for $\ln R$ against $\ln m$ is $\ln R=0.74\ln m+3.2$. Write down $b$ and find $k$.
   (iii) Use this model to predict the metabolic rate of a mammal of mass $70$ kg.

   [6 marks]

   (b)

   For mammals of one species, $k=12$ and $b=0.75$. The metabolic rate of a $16$ kg mammal is doubled. Find the mass of a mammal of this species with this doubled metabolic rate.

   [6 marks]

   Total for question 4: 12 marks

End of questions