document.write( "Question 724106: Write as a single logarithm :\r
\n" ); document.write( "\n" ); document.write( "1/2 log3 x + log3 y -4 log3 z \n" ); document.write( "
Algebra.Com's Answer #443689 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"%281%2F2%29log%283%2C+%28x%29%29+%2B+log%283%2C+%28y%29%29+-+4log%283%2C+%28z%29%29\"
\n" ); document.write( "Like logarithmic terms have the same bases and same arguments. These logs have the same base, 3, but their arguments are all different. So they are not like terms and cannot just be added and subtracted.

\n" ); document.write( "But there is another way to combine these terms. Two properties of logarithms, \"log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29\" and \"log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29\", allow us to combine two logs if their bases are the same and their coefficients are 1's. The first and third logs have coefficients of 1/2 and 4 so it appears that we cannot use these properties either.

\n" ); document.write( "Fortunately there is another property of logarithms, \"log%28a%2C+%28p%5En%29%29+=+n%2Alog%28a%2C+%28p%29%29\", which allows us to move the coefficient. This property, reading it from right to left, lets us move a coefficient into the argument as its exponent. So we can use this property to move the coefficients out of the way:
\n" ); document.write( "\"log%283%2C+%28x%5E%281%2F2%29%29%29+%2B+log%283%2C+%28y%29%29+-+log%283%2C+%28z%5E4%29%29\"
\n" ); document.write( "Since an exponent of 1/2 means square root, I'm going to rewrite the first argument as a square root:
\n" ); document.write( "\"log%283%2C+%28sqrt%28x%29%29%29+%2B+log%283%2C+%28y%29%29+-+log%283%2C+%28z%5E4%29%29\"

\n" ); document.write( "Now that the coefficients are all 1's we can start using the other properties to combine them. Starting with the first two logs, we will use the first property because it logs, like ours, have a \"+\" between them:
\n" ); document.write( "\"log%283%2C+%28sqrt%28x%29%2Ay%29%29+-+log%283%2C+%28z%5E4%29%29\"
\n" ); document.write( "With the last two logs, we will use the second property because its logs, like ours, have a \"-\" between them:
\n" ); document.write( "\"log%283%2C+%28%28sqrt%28x%29%2Ay%29%2Fz%5E4%29%29\" \n" ); document.write( "
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