Question

The decibel rating D| is related to the sound intensity I| by the formula D = 10 log_10 (I/10^-16)| for the noise level in decibels. Let D| and d| represent the decibel ratings of sounds of intensity I| and i|, respectively. Using properties of logarithms, find a simplified formula for the difference between the two ratings, D-d|, in terms of the two intensities I land i|. help (logarithms) (Enter log10 or logten for the base 10 logarithm.) If a sound's intensity quadruples, how many decibels louder does the sound become? decibels help (numbers)

Answer 1

Soluiton

             The Decibel rating   D   is   related to   sound intensity I by   the formula

              $D = 10log_{10}\left ( \frac{I}{10^{-16}} \right )$

      a) Let D and    d represent the decibel ratings of sounds of intensity

             I     and    i    respectively

            Using   Properties   of    logarithms , find a   simplified   formula   for   the difference

            between the two ratings D   and   d    in   terms   of    I    and    i

            We have ,

                   $D = 10log_{10}\left ( \frac{I}{10^{-16}} \right )$              and

                   $d = 10log_{10}\left ( \frac{i}{10^{-16}} \right )$

                Now , D - d

                     $D - d = 10log_{10}\left ( \frac{I}{10^{-16}} \right )- 10log_{10}\left ( \frac{i}{10^{-16}} \right )$

                   $D - d = 10 \left [ log_{10}\left ( \frac{I}{10^{-16}} \right ) - log_{10}\left ( \frac{i}{10^{-16}} \right ) \right ]$

   $D - d = 10 \left [ log_{10}I - log_{10}10^{-16} - log_{10}i +log_{10}10^{-16}\right ]$   

                     $D - d = 10 \left [ log_{10}I - log_{10}i \right ]$

                     $D - d = 10log_{10}\left ( \frac{I}{i} \right )$

       

b) If     a     Sound's intensity quadruples , To find how many decibels louder

does the    sound become

            From    the above    Part , We have

$D = 10log_{10}\left ( \frac{I}{10^{-16}} \right )$   

              Here , We   have    I = 4

              $D = 10log_{10}\left ( \frac{4}{10^{-16}} \right )$

   $D = 10log_{10}4 - 10log_{10}(10^{-16})$

               $D = 10log_{10}4 +160log_{10}10$

              $D = 10log_{10}4 +160$

                $D = 10\times 0.6020+160$

                $D = 6.020+160$

                 $D = 166.020$