document.write( "Question 1103106: Given that log(base a) x = p and log(base a) y = q, find the following: 1) log(base xy) a = ? and 2) log(base y) (ax^(2)y^(3)) = ?\r
\n" ); document.write( "\n" ); document.write( "I've been staring at this for a good twenty minutes and I cannot for the life of me figure it out. Any help is much appreciated. Thanks! \n" ); document.write( "

Algebra.Com's Answer #717811 by KMST(5328) $\"\"$ $\"About$

You can put this solution on YOUR website!
The hardest to understand (and remember) properties of logarithms are
\n" ); document.write( "flipping base and number: $\"log%28Y%2CB%29\"$ $\"%22=%22\"$ $\"1%2Flog%28B%2CY%29\"$
\n" ); document.write( "and
\n" ); document.write( "change of base: $\"log%28Y%2CX%29\"$ $\"%22=%22\"$ $\"log%28B%2CX%29%2Flog%28B%2CY%29\"$ .
\n" ); document.write( "If you are going to \"apply properties of logarithms\" you need those two here.
\n" ); document.write( "You also need to know that $\"log%28B%2CXY%29=log%28B%2CX%29%2Blog%28B%2CY%29\"$ and that
\n" ); document.write( " $\"log%28B%2CX%5En%29=n%2Alog%28B%2CX%29\"$ .
\n" ); document.write( "
\n" ); document.write( "1) If you do not want to (or cannot) memorize or remember properties, you do not need to.
\n" ); document.write( "It all flows logically from the meaning of logarithm.
\n" ); document.write( " $\"log%28a%2Cx%29=p\"$ means that $\"a%5Ep=x\"$ , and
\n" ); document.write( " $\"log%28a%2Cy%29=q\"$ means that $\"a%5Eq=y\"$ .
\n" ); document.write( "That means that $\"a%5E%28p%2Bq%29=a%5Ep%2Aa%5Eq=xy\"$ ,
\n" ); document.write( "and if $\"xy=a%5E%28p%2Bq%29\"$ .
\n" ); document.write( "Then, you understand that
\n" ); document.write( " $\"xy%5E%221+%2F+%28+p%2Bq+%29%22\"$ $\"%22=%22\"$ $\"%28a%5E%28p%2Bq%29%29%5E%281%2F%28p%2Bq%29%29\"$ $\"%22=%22\"$ $\"a%5E%22%28p%2Bq%29+%2F+%28p%2Bq%29%22\"$ $\"%22=%22\"$ $\"a%5E1=a\"$ ,
\n" ); document.write( "and the fact that $\"xy%5E%221+%2F+%28+p%2Bq+%29%22=a\"$ means that
\n" ); document.write( " $\"log%28xy%2Ca%29\"$ $\"%22=%22\"$ $\"highlight%281%2F%28p%2Bq%29%29\"$ .
\n" ); document.write( "
\n" ); document.write( "INVOKING PROPERTIES OF LOGARITHMS:
\n" ); document.write( "1)
\n" ); document.write( " $\"log%28xy%2Ca%29\"$ $\"%22=%22\"$ $\"1%2Flog%28a%2Cxy%29\"$ $\"%22=%22\"$ $\"1%2F%28log%28a%2Cx%29%2Blog%28a%2Cy%29%29\"$ $\"%22=%22\"$ $\"highlight%281%2F%28p%2Bq%29%29\"$ .
\n" ); document.write( "
\n" ); document.write( "2)
\n" ); document.write( " $\"log%28y%2Cax%5E2y%5E3%29\"$ $\"%22=%22\"$ $\"log%28y%2Ca%29\"$ $\"%22%2B%22\"$ $\"log%28y%2Cx%5E2%29\"$ $\"%22%2B%22\"$ $\"log%28y%2Cy%5E3%29\"$ $\"%22=%22\"$ $\"log%28y%2Ca%29\"$ $\"%22%2B%22\"$ $\"2log%28y%2Cx%29\"$ $\"%22%2B%22\"$ $\"3log%28y%2Cy%29\"$ $\"%22=%22\"$ $\"1%2Flog%28a%2Cy%29\"$ $\"%22%2B%22\"$ $\"2log%28a%2Cx%29%2Flog%28a%2Cy%29\"$ $\"%22%2B%22\"$ $\"3%2A1\"$ $\"%22=%22\"$ $\"highlight%281%2Fq%2B2p%2Fq%2B3%29\"$ \r
\n" ); document.write( "\n" ); document.write( "1) If you cannot memorize or remember properties, you do not need to.
\n" ); document.write( "It all flows logically from the meaning of logarithm.
\n" ); document.write( " $\"log%28a%2Cx%29=p\"$ means that $\"a%5Ep=x\"$ , and
\n" ); document.write( " $\"log%28a%2Cy%29=q\"$ means that $\"a%5Eq=y\"$ .
\n" ); document.write( "That means that $\"a%5E%28p%2Bq%29=a%5Ep%2Aa%5Eq=xy\"$ ,
\n" ); document.write( "and if $\"xy=a%5E%28p%2Bq%29\"$ .
\n" ); document.write( "Then, you understand that
\n" ); document.write( " $\"xy%5E%221+%2F+%28+p%2Bq+%29%22\"$ $\"%22=%22\"$ $\"%28a%5E%28p%2Bq%29%29%5E%281%2F%28p%2Bq%29%29\"$ $\"%22=%22\"$ $\"a%5E%22%28p%2Bq%29+%2F+%28p%2Bq%29%22\"$ $\"%22=%22\"$ $\"a%5E1=a\"$ ,
\n" ); document.write( "and the fact that $\"xy%5E%221+%2F+%28+p%2Bq+%29%22=a\"$ means that
\n" ); document.write( " $\"log%28xy%2Ca%29\"$ $\"%22=%22\"$ $\"highlight%281%2F%28p%2Bq%29%29\"$ .\r
\n" ); document.write( "\n" ); document.write( "NOTE:
\n" ); document.write( "The \"properties of logarithms\" sound like something to memorize,
\n" ); document.write( "and something to be anxious about,
\n" ); document.write( "but there is no need to be scared.
\n" ); document.write( " $\"log%28a%2Cx%29=p\"$ is read as
\n" ); document.write( "the logarithm on base $\"a\"$ of $\"x\"$ is $\"p\"$ .
\n" ); document.write( "The exponent on base $\"a\"$ to get $\"x\"$ is $\"p\"$ sounds easier,
\n" ); document.write( "by just replacing the word \"exponent\" for the scary word \"logarithm.\" \n" ); document.write( "

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