Chapter 2.6, Problem 87E

Sound intensity: decibels. Because of the extraordinary range of sensitivity of the human ear (a range of over $1,000$ million millions to $1$ ), it is helpful to use a logarithmic scale, rather than an absolute scale, to measure sound intensity over this range. The unit of measure is called the decibel, after the inventor of the telephone, Alexander Graham Bell. If we let $N$ be the number of decibels, $I$ the power of the sound in question (in watts per square centimeter), and $I_0$ the power of sound just below the threshold of hearing (approximately ${10}^{-16}$ watt per square centimeter), then

$I=I_0{10}^{N/10}$

Show that this formula can be written in the form

$N=10\log \frac{I}{I_0}$