document.write( "Question 633521: Rewrite th expression as a single logarithm.
\n" ); document.write( "log(3)5+5log(3)2 \n" ); document.write( "
Algebra.Com's Answer #399070 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
Logarithms that are not base 10 or base e logarithms are difficult to post clearly. I recommend that you try using some English, like:
\n" ); document.write( "\"(base 3 log of 5) + 5*(base 3 log of 2)\"
\n" ); document.write( "or try to learn the syntax of Algebra.com's formula drawing software. Click on the \"Show source\" link above to see what I typed to get yoru expreesion to look like:
\n" ); document.write( "\"log%283%2C+%285%29%29+%2B+5log%283%2C+%282%29%29\"
\n" ); document.write( "Tutors are less likely to respond when they are not sure what the problem actually is. So please try to be clearer when you post your logarithms.

\n" ); document.write( " We have three logarithms. We want only one. Somehow we must find a way to condense/combine the three into one.

\n" ); document.write( "As the expression above shows, I'm guessing that your expression has base 3 logarithms. If this is not correct then you will have to re-post your problem.

\n" ); document.write( "Your expression has two logarithms and the problem is to rewrite this as a single logarithm. Somehow we must find a way to condense/combine the two logs into one.

\n" ); document.write( "There are two ways to combine logarithmic terms:
  • Add or subtract them. As usual, however, the terms must be like terms. Like logarithmic terms have the same bases and the same arguments.
  • Use either of the following properties:
    • \"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\"
    These properties require that the bases be the same and the coefficients be 1's.
Your logarithms all have the same base, 3. But the arguments are different so we cannot add or subtract them. The coefficients are not 1's either so we cannot use the properties... at least not yet. Fortunately there is another property of logarithms, \"q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29\", which allows us to \"move\" a coefficient into the argument as its exponent. With this third property we can get the coefficients of 1's we need for the other properties.

\n" ); document.write( "So we start by using the third property on the second log to move the 5 into the argument as its exponent:
\n" ); document.write( "\"log%283%2C+%285%29%29+%2B+log%283%2C+%282%5E5%29%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"log%283%2C+%285%29%29+%2B+log%283%2C+%2832%29%29\"

\n" ); document.write( "Now the two logs have coefficients of 1. So we can use the first two properties to combine them. We will use the first property be cause its logs, like ours, have a \"+\" between them.
\n" ); document.write( "\"log%283%2C+%285%2A32%29%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"log%283%2C+%28160%29%29\" \n" ); document.write( "
\n" );