document.write( "Question 269655: Express as a single logarithm,and, if possible, simplify\r
\n" ); document.write( "\n" ); document.write( " Logb5x+7(logbx – logby) \n" ); document.write( "
Algebra.Com's Answer #197955 by jsmallt9(3758)\"\" \"About 
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Taking an expression of multiple logarithms as a single logarithm requires that we know the properties of logarithms:
  • \"log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29\"
  • \"log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29\"
  • \"q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29\"

\n" ); document.write( "The first two are used to combine two logarithms into one. The first one is use when there is a \"+\" between the two logs. The second one is used when there is a \"-\" between the two logs. The first two properties require that the two logarithms have coefficients of 1 in front of them. The third property is used to move other coefficients into the argument of the logarithm. Let's see how this works on your expression:
\n" ); document.write( "\"log%28b%2C+%285x%29%29+%2B+7%28log%28b%2C+%28x%29%29+-+log%28b%2C+%28y%29%29%29\"
\n" ); document.write( "We'll start by combining the two logarithms in the parentheses using the second property:
\n" ); document.write( "\"log%28b%2C+%285x%29%29+%2B+7%28log%28b%2C+%28x%2Fy%29%29%29\"
\n" ); document.write( "or
\n" ); document.write( "\"log%28b%2C+%285x%29%29+%2B+7%2Alog%28b%2C+%28x%2Fy%29%29\"
\n" ); document.write( "Before we combine the last two logarithms we need to have coefficients of 1. So that 7 needs to go. This is where the thrid property comes in handy:
\n" ); document.write( "\"log%28b%2C+%285x%29%29+%2B+log%28b%2C+%28%28x%2Fy%29%5E7%29%29\"
\n" ); document.write( "or
\n" ); document.write( "\"log%28b%2C+%285x%29%29+%2B+log%28b%2C+%28x%5E7%2Fy%5E7%29%29\"
\n" ); document.write( "Now we can combine the last two logarithms (with the first property):
\n" ); document.write( "\"log%28b%2C+%285x%2A%28x%5E7%2Fy%5E7%29%29%29\"
\n" ); document.write( "which simplifies to
\n" ); document.write( "\"log%28b%2C+%285x%5E8%2Fy%5E7%29%29\" \n" ); document.write( "
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