document.write( "Question 814155: Write as a single logarithm:
\n" ); document.write( "(1/2) log7 W - (2/3) log7 X + (3/4) log7 Y - (4/5) log7 Z \n" ); document.write( "

Algebra.Com's Answer #490170 by jsmallt9(3758) $\"\"$ $\"About$

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The following properties of logarithms are often used in problems like this:

$\"log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29\"$
$\"log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29\"$
$\"n%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5En%29%29\"$

The first two will combine logs into a single log, the first for when there is a \"+\" between the logs and the second for when there is a \"-\" between. These two properties require that the bases of the logs be the same and that the coefficients of the logs are 1's. The third property can be used to \"move\" a coefficient that is not 1 into the argument as its exponent.

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\n" ); document.write( "Since none of the logs in your expression have a coefficient of 1, we will start by using the third property to move those coefficients into the argument as the exponent:
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\n" ); document.write( "All those fractional exponents represent roots of various kinds. Since radicals display better on algebra.com I am going to rewrite all those roots in radical form before proceeding:
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\n" ); document.write( "Now we can start using the other two properties to combine these logs. The first two have a \"-\" between them so we use the second property:
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\n" ); document.write( "The first two logs of what remains have a \"+\" between them so we use the first property:
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\n" ); document.write( "The remaining two logs have a \"-\" between them so back to the second property:
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\n" ); document.write( "which simplifies to:
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