- Identify the domain of a logarithmic function.
- Graph logarithmic functions.
Logarithmic Functions in Sound Engineering
Sound engineers use logarithmic functions to model sound intensity, frequency response, and acoustic properties. Graphing logarithmic transformations helps visualize sound distribution patterns and optimize audio equipment placement for the best audience experience.
Why Logarithms are Used in Sound
Human hearing spans an enormous range—from the faintest audible whisper ([latex]10^-12 W/m^2[/latex]) to the threshold of pain ([latex]1 W/m^2[/latex]). The decibal (dB) scale compresses this range using a base-10 logarithm:
[latex]L=10log_10(\frac{I}{I_0})[/latex]
where [latex]I[/latex] is sound intensity and [latex]I_0=10^-12 W/m^2[/latex] is the reference intensity.
The parent function [latex]f(x) = \log_{10}(x)[/latex]
has the following characteristics:
- Domain: [latex](0,\infty)[/latex]
- Range: [latex](-\infty,\infty)[/latex]]
- Vertical asymptote: [latex]x=0[/latex]
Compare these three functions:
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[latex]f_1(x) = 2\log_{10}(x+1) - 2[/latex]
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[latex]f_2(x) = -2\log_{10}(x+1) - 2[/latex]
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[latex]f_3(x) = 2\log_{10}(-x-1) - 2[/latex]
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Identify the domain, asymptote, and vertical shift for each.
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Graph all three on the same coordinate plane.
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Which two curves could represent real-world EQ settings? Which one is only a math exploration?
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Explain your reasoning in terms of audio.