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IB Maths AI SL 1.5 Integer Exponents and Logarithms

IB Maths AI SL 1.5 Integer Exponents and Logarithms
IB Mathematics: applications and interpretation guide, first assessment 2021

Practise applying exponent and logarithm laws to rearrange and solve expressions, including pH, sound intensity and other contextual scales written in standard form.

How this is tested

  • rewrite between exponential and logarithmic form before isolating the required quantity
  • apply log product, quotient and power laws while keeping every argument in its valid domain
  • substitute into pH or decibel models and express very large or small results in standard form

Question 7

[Maximum number: 5]

The loudness of a sound, L, measured in decibels (dB) is determined by the intensity of the sound, I, measured in watts per square metre (Wm−2)\left(\mathrm{Wm}^{-2}\right). The relationship between loudness and intensity can be expressed using the logarithmic function

L=10log⁡10(II0),I>0L=10 \log _{10}\left(\frac{I}{I_{0}}\right), I>0

where I0I_{0} is the reference intensity (the intensity of the least audible sound to the human ear).
The reference intensity I0I_{0} is 10−12Wm−210^{-12} \mathrm{Wm}^{-2}.
The intensity of sound on a busy street is 10−5Wm−210^{-5} \mathrm{Wm}^{-2}.

Question 7(a)

(a)

Calculate the loudness of the sound.

The sound of a jet engine reaches a loudness of 185 dB .

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Question 7(b)

(b)

Determine the intensity of its sound. Give your answer in the form a×10ka \times 10^{k} where 1≤a<10,k∈Z1 \leq a<10, k \in \mathbb{Z}.

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