document.write( "Question 818477: Write the expression as a single log?\r
\n" ); document.write( "\n" ); document.write( "21log3 (cuberoot(x)) + log3 (9x^2) - log5 25\r
\n" ); document.write( "\n" ); document.write( "Thanks! \n" ); document.write( "
Algebra.Com's Answer #492606 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
I'm assuming that the numbers immediately after \"log\" are supposed to be the bases of these logs. If I am wrong, then you will have to re-post your question. In the future, either use some English to describe your logs (like \"base 5 log of (25) for \"log%285%2C+%2825%29%29\", or teach yourself algebra.com's syntax for formulas. (Click on the \"Show source\" link above to see what I typed to get that logarithm to display so nicely.)

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\n" ); document.write( "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 are used to combine logs. They require that the logs have the same bases and coefficients of 1. The third property, since it allows us to \"move\" a coefficient out of the way, is often used to get coefficients of 1 so the other two properties may be used.

\n" ); document.write( "The first log has a coefficient that is not a 1, it is 21. So we will use the third property to move the 21 into the argument as its exponent:
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\n" ); document.write( "We can use fractional exponents to simplify the first argument. Since a cube root is the same as an exponent of 1/3:
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\n" ); document.write( "The rule for powers of a power is to multiply. Since 1/3*21 = 7:
\n" ); document.write( "\"log%283%2C+%28x%5E7%29%29+%2B+log%283%2C+%289x%5E2%29%29+-+log%285%2C+%2825%29%29\"

\n" ); document.write( "Now we can use the first property to combine the first two logs:
\n" ); document.write( "\"log%283%2C+%28%28x%5E7%29%2A%289x%5E2%29%29%29+-+log%285%2C+%2825%29%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"log%283%2C+%289x%5E9%29%29+-+log%285%2C+%2825%29%29\"

\n" ); document.write( "We cannot use the second property to combine the remaining logs because the bases are different, 3 and 5. But the second log is one we can figure out \"by hand\". \"log%285%2C+%2825%29%29\" represents the exponent one would put on a 5 to get a result of 25. We know what this exponent is. It is 2. So \"log%285%2C+%2825%29%29+=+2\". This makes our expression:
\n" ); document.write( "\"log%283%2C+%289x%5E9%29%29+-+2\"
\n" ); document.write( "This might be an acceptable answer. It is an expression with a single logarithm. But the problem, as you posted it, says \"express as a single log\" not \"express with a single log\" like we have above. If the final answer is really supposed to be just a single log and nothing else, then we have some more work to do.

\n" ); document.write( "Here we can use a little trick. We are going to turn the \"2\" into a logarithm with a base of 3. This will allow us to combine the remaining terms into a single logarithm. \"log%283%2C+%283%29%29+=+1\" since the exponent for 3 that results in a 3 is 1. So I will multiply the 2 by (a weird-looking) 1:
\n" ); document.write( "\"log%283%2C+%289x%5E9%29%29+-+2%2Alog%283%2C+%283%29%29\"
\n" ); document.write( "Now we use the third property to get the coefficient of 2 out of the way:
\n" ); document.write( "\"log%283%2C+%289x%5E9%29%29+-+log%283%2C+%28%283%29%5E2%29%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"log%283%2C+%289x%5E9%29%29+-+log%283%2C+%289%29%29\"
\n" ); document.write( "Now we can use the second property to combine the terms:
\n" ); document.write( "\"log%283%2C+%28%289x%5E9%29%2F9%29%29\"
\n" ); document.write( "The factors of 9 cancel:
\n" ); document.write( "\"log%283%2C+%28x%5E9%29%29\" \n" ); document.write( "
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