Model and solve problems that involve logarithmic functions, e.g. decibels in acoustics and the Richter scale for earthquake magnitude, with and without technology.

The sound intensity level in decibels (dB) at a location is given by the formula $$L = 10 \log_{10}\left(\frac{I}{I_0}\right)$$ where \(L\) is the intensity level in decibels, \(I\) is the sound intensity in watts per square metre (W/m²), and \(I_0 = 10^{-12}\) W/m² is the reference intensity. (a) A whisper has a sound intensity level of 30 dB. Calculate the sound intensity \(I\) of the whisper. [2 marks] (b) Hence, find how many times more intense the whisper is than the reference intensity \(I_0\). [1 mark]

The loudness level of a sound, measured in decibels (dB), can be modelled by the logarithmic function \( L = 10\log_{10}\left(\frac{P}{P_0}\right) \), where \( L \) is the loudness level in decibels, \( P \) is the sound power in watts per square metre, and \( P_0 \) is a reference sound power (the threshold of human hearing). A chainsaw produces a loudness level of \( 110 \) dB. A vacuum cleaner produces a loudness level of \( 75 \) dB. (a) Determine a simplified logarithmic equation for the difference in loudness levels between the chainsaw and the vacuum cleaner in terms of their sound powers. (1 mark) (b) Convert your equation from part (a) to index form, making the ratio of sound powers the subject. (1 mark) (c) Calculate how many times more powerful the sound from the chainsaw is compared to the sound from the vacuum cleaner. Give your answer correct to the nearest whole number. (2 marks)

How many times more intense is the sound at location A than the sound at location B?

The loudness of sound is measured in decibels (dB) using the logarithmic model \(L = 10\log_{10}\left(\frac{I}{I_0}\right)\), where \(L\) is the loudness in decibels, \(I\) is the intensity of the sound in watts per square metre (\(\text{W/m}^2\)), and \(I_0\) is the reference intensity. Refer to the table below showing sound intensity measurements at three different locations in a concert venue. (a) Determine a simplified logarithmic equation for the difference in loudness between any two locations in terms of their intensities. (2 marks) (b) Calculate the difference in loudness between Location X and Location Y. Give your answer correct to one decimal place. (2 marks) (c) Hence determine how many times more intense the sound is at Location Z compared with Location Y. (1 mark)