document.write( "Question 815145: A law of physics states that the intensity of sound is inversely proportional to the square of the distance d of the source I: k/d^2\r
\n" ); document.write( "\n" ); document.write( "a. Use this model and the equation B=10log(I/I0) to show that the decibel levels B1 and B2 at distances d1 and d2 from a sound source are related by the equation B1= B2-20log(d1/d2) \r
\n" ); document.write( "\n" ); document.write( "I'm a little confused. Help?
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #490743 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
Since \"B=10log%28%28I%2FI%5B0%5D%29%29\" then
\n" ); document.write( "\"B%5B1%5D=10log%28%28I%5B1%5D%2FI%5B0%5D%29%29\" and \"B%5B2%5D=10log%28%28I%5B2%5D%2FI%5B0%5D%29%29\"

\n" ); document.write( "We want \"B%5B1%5D\" and \"B%5B2%5D\" in the same equation. So we will combine the two above by adding or subtracting them. I can see ahead that subtraction will make things a little easier so we will subtract the second equation from the first one:
\n" ); document.write( "\"B%5B1%5D-B%5B2%5D=10log%28%28I%5B1%5D%2FI%5B0%5D%29%29-10log%28%28I%5B2%5D%2FI%5B0%5D%29%29\"
\n" ); document.write( "Adding \"B%5B2%5D\" to each side:
\n" ); document.write( "
\n" ); document.write( "As you can, the desired equation is starting to appear. We just need to manipulate the right side so that
  • the I's are gone
  • The d's are there
  • And it looks like the desired equation. The I's and d's are related by the equation: \"I+=+k%2Fd%5E2\". So \"I%5B1%5D+=+k%2Fd%5B1%5D%5E2\" and \"I%5B2%5D+=+k%2Fd%5B2%5D%5E2\". Substituting for \"I%5B1%5D\" and \"I%5B2%5D\" we get:
    \n" ); document.write( "
    \n" ); document.write( "Most of the I's are gone and we have the d's. All that is left is to eliminate the \"I%5B0%5D\"'s and the k's and make the equation look like the one we want. We do want only one log so I combine the two we have next. Factoring out 10:
    \n" ); document.write( "
    \n" ); document.write( "Now we can use the \"log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29\" property to combine the logs:
    \n" ); document.write( "
    \n" ); document.write( "Rewriting the division of fractions as a multiplication by the reciprocal:
    \n" ); document.write( "
    \n" ); document.write( "...and the \"I%5B0%5D\"'s and the k's cancel out!
    \n" ); document.write( "
    \n" ); document.write( "which simplifies to:
    \n" ); document.write( "\"B%5B1%5D=B%5B2%5D%2B10%28log%28%28d%5B2%5D%5E2%2Fd%5B1%5D%5E2%29%29%29\"
    \n" ); document.write( "We're getting very close. Rewriting the fraction:
    \n" ); document.write( "\"B%5B1%5D=B%5B2%5D%2B10%28log%28%28%28d%5B2%5D%2Fd%5B1%5D%29%5E2%29%29%29\"
    \n" ); document.write( "Using the \"log%28a%2C+%28p%5En%29%29+=+n%2Alog%28a%2C+%28p%29%29\" property:
    \n" ); document.write( "\"B%5B1%5D=B%5B2%5D%2B10%282log%28%28d%5B2%5D%2Fd%5B1%5D%29%29%29\"
    \n" ); document.write( "Multiplying the 10 and 2:
    \n" ); document.write( "\"B%5B1%5D=B%5B2%5D%2B20log%28%28d%5B2%5D%2Fd%5B1%5D%29%29\" \n" ); document.write( "
\n" );