document.write( "Question 365314: Write the expression as a single logarithm with a coefficient of 1.
\n" ); document.write( "logb(8) + 5[logb(7 + x) − 1/2 logb(6 − x)] \n" ); document.write( "
Algebra.Com's Answer #260546 by jsmallt9(3758)\"\" \"About 
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\n" ); document.write( "There are three properties of logarithms we will need for this problem:
  • \"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( "All of these properties can be used in both directions. But for this problem we will using them from left to right. The third property allows us to move a coefficient into the argument as an exponent. We will use it to move the (1/2) in front of the last log into its argument:
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\n" ); document.write( "Since 1/2 as an exponent means square root, I'll replace the last argument with a square root:
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\n" ); document.write( "Next, inside the parentheses we have two logarithms of the same base with coefficients of 1 and with a \"-\" between them. This is a job for the second property:
\n" ); document.write( "\"log%28b%2C+%288%29%29+%2B+5%28log%28b%2C+%28%287+%2B+x%29%2Fsqrt%286+-+x%29%29%29%29\"
\n" ); document.write( "or
\n" ); document.write( "\"log%28b%2C+%288%29%29+%2B+5%2Alog%28b%2C+%28%287+%2B+x%29%2Fsqrt%286+-+x%29%29%29\"
\n" ); document.write( "Next we can use the third property again to move the 5 from in front of the second log:
\n" ); document.write( "\"log%28b%2C+%288%29%29+%2B+log%28b%2C+%28%28%287+%2B+x%29%2Fsqrt%286+-+x%29%29%5E5%29%29\"
\n" ); document.write( "And last, with two logs of the same base with coefficients of 1 and a \"+\" between them, we can use the first property to combine them:
\n" ); document.write( "\"log%28b%2C+%288%2A%28%287+%2B+x%29%2Fsqrt%286+-+x%29%29%5E5%29%29\"
\n" ); document.write( "which is a single logarithm with a coefficient of 1. (NOTE: You do have a square root in a denominator. So you may need to rationalize the denominator. I'll leave that up to you.) \n" ); document.write( "
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