document.write( "Question 411048: expand each logarithiam \r
\n" ); document.write( "\n" ); document.write( "log4 a^4/b^3\r
\n" ); document.write( "\n" ); document.write( "log3 sqrt 2x\r
\n" ); document.write( "\n" ); document.write( "log3 x^2y^3 \n" ); document.write( "
Algebra.Com's Answer #289258 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
Please post 1 (maybe 2) problems in each post. Posting three problems may be the reason that you have not received a response prior to this. I will do one and perhaps that will help you figure out the others. (If not, then repost them.)

\n" ); document.write( "There are three properties of logarithms that are used when \"expanding\" logarithms:
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  • \"log%28a%2C+%28p%2Aq%29%29+=+log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29\" which shows us how to write the log of a product as the sum of the logs of the factors (and vice versa).
  • \"log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29\" which shows us how to write the log of a quotient as the difference of the logs of the numerator and denominator (and vice versa).
  • \"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\" which shows us how to write the log of a power as the exponent times the log of the base of the exponential argument (and vice versa).

\n" ); document.write( "\"log%284%2C+%28a%5E4%2Fb%5E3%29%29\"
\n" ); document.write( "The argument is a quotient. So we will use the second property to get:
\n" ); document.write( "\"log%284%2C+%28a%5E4%29%29+-+log%284%2C+%28b%5E3%29%29\"
\n" ); document.write( "Both of these logs are logs of a power. So we will use the third property on both to get:
\n" ); document.write( "\"4%2Alog%284%2C+%28a%29%29+-+3%2Alog%284%2C+%28b%29%29\"
\n" ); document.write( "When the arguments are no longer a product, quotient or power, then the expression is fully \"expanded\". So we are finished.

\n" ); document.write( "Some hints for \"log%283%2C+%28sqrt%282a%29%29%29\":
  • Use the fact that square roots are exponents of 1/2 to rewrite \"sqrt%282a%29\" as \"%282a%29%5E%281%2F2%29\".
  • Eventually you will be using the first property on \"log%283%2C+%282a%29%29\". When you replace this with what the first property shows you, put the new expression in parentheses when you substitute. I think you'll see why once you do this.
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