In Exercises 85-88, use the following information. The...

In Exercises $85-88,$ use the following information. The relationship between the number of decibels $\beta$ and the intensity of a sound I in watts per square meter is given by $$ \boldsymbol{\beta}=10 \log \left(\frac{I}{10^{-12}}\right) $$ Use the properties of logarithms to write the formula in simpler form, and determine the number of decibels of a sound with an intensity of $10^{-6}$ watt per square meter.

In Exercises $85-88,$ use the following information.
The relationship between the number of decibels $\beta$ and
the intensity of a
sound I in watts
per square meter is
given by
$$
\boldsymbol{\beta}=10 \log \left(\frac{I}{10^{-12}}\right)
$$
Use the properties of logarithms to write the
formula in simpler form, and determine the
number of decibels of a sound with an intensity
of $10^{-6}$ watt per square meter.

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Key Concepts

Sound Intensity and the Decibel Scale

The decibel scale is a logarithmic measure used to quantify the intensity of a sound relative to a reference level. This scale compresses a wide range of sound intensities into more manageable numbers, aligning with the human ear’s nonlinear perception of sound. The decibel formula, which involves the logarithm of the ratio of the sound intensity to a reference intensity, is fundamental in fields such as acoustical engineering and audio technology.

Properties of Logarithms

Logarithmic properties, such as the product, quotient, and power rules, are essential tools for simplifying complex logarithmic expressions. By applying rules like log(a/b) = log(a) - log(b) and log(a^n) = n log(a), expressions that involve ratios or powers can be rewritten into simpler, more manageable forms. This approach is widely used in various fields, including exponential growth and decay, acoustics, and signal processing.

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