document.write( "Question 594905: express 1/2lnx-4ln(x-1)+3ln(x+7) as a single logarithm \n" ); document.write( "
Algebra.Com's Answer #376953 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"%281%2F2%29ln%28x%29-4ln%28x-1%29%2B3ln%28x%2B7%29\"
\n" ); document.write( "To write this expression as a single logarithm, we will have to combine the three terms into one. There are two ways to combine logarithmic terms:
  • Actually adding and/or subtracting them. This requires that the terms be like terms. Like logarithmic terms have the same bases and the same arguments.
  • Use one or both of the following 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\"

\n" ); document.write( "The logs in your expression all have the same base but the arguments are different. Since there is no way to make the arguments the same, you will not be able to add or subtract them.

\n" ); document.write( "The properties require the same base and coefficients of 1. Your logs have the same base but the none of the coefficients are 1's. So it might seem that these properties can't be used. But fortunately there is another property of logarithms, \"q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29\", which allows one to \"move\" a coefficient into the argument as its exponent (and vice versa for that matter). So with this third property we will be able to move the coefficients out of the way!

\n" ); document.write( "So we start by moving the coefficients, using the third property mentioned above:
\n" ); document.write( "\"ln%28x%5E%281%2F2%29%29-ln%28%28x-1%29%5E4%29%2Bln%28%28x%2B7%29%5E3%29\"
\n" ); document.write( "(Since 1/2 as an exponent means square root, I'm going to write it as a square root.)
\n" ); document.write( "\"ln%28sqrt%28x%29%29-ln%28%28x-1%29%5E4%29%2Bln%28%28x%2B7%29%5E3%29\"

\n" ); document.write( "Now that the coefficients are all ones, we can start combining terms. With the \"-\" between the two terms we will use the second property (which also has a \"-\" between the terms) to combine them into one:
\n" ); document.write( "\"ln%28sqrt%28x%29%2F%28x-1%29%5E4%29%2Bln%28%28x%2B7%29%5E3%29\"

\n" ); document.write( "Now we can combine the last two terms. Since there is a \"+\" between the terms, we will use the first property (which also has a \"+\" between the terms):
\n" ); document.write( "\"ln%28%28sqrt%28x%29%2F%28x-1%29%5E4%29%2A%28x%2B7%29%5E3%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"ln%28%28sqrt%28x%29%2A%28x%2B7%29%5E3%29%2F%28x-1%29%5E4%29\"

\n" ); document.write( "We have the desired single logarithm and the fraction cannot be reduced. So we are finished. \n" ); document.write( "
\n" );