Chapter 1.6, Problem 91E

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 n umber of decibels, I the power of the sound in question (in watts per square centimeter), and I₀ 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

  $N=10\log \frac{I}{I_0}$