document.write( "Question 275180: Can you help me write $\"log%28a%2C%2840%2F%28a%5E3%29%29%29\"$ in terms of x and y if $\"log%28a%2C2%29=x\"$ , and $\"log%28a%2C25%29=y\"$ ? Thanks! \n" ); document.write( "

Algebra.Com's Answer #200816 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"log%28a%2C+%2840%2F%28a%5E3%29%29%29\"$
\n" ); document.write( "First let's split this logarithm into two. We can use the property of logarithms, $\"log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29\"$ , to do this:
\n" ); document.write( " $\"log%28a%2C+%2840%29%29+-+log%28a%2C+%28a%5E3%29%29\"$
\n" ); document.write( "The second logarithm simplifies:
\n" ); document.write( " $\"log%28a%2C+%2840%29%29+-+3\"$
\n" ); document.write( "Now we need to express the remaining logarithm in terms of x and y. If we factor 40 we might be able to see a path to our solution:
\n" ); document.write( " $\"log%28a%2C+%282%2A2%2A2%2A5%29%29+-+3\"$
\n" ); document.write( "If we can separate the 2's into separate logarithms, each of them would be an \"x\". We can use another property of logarithms,

\n" ); document.write( "The first three logs are x:
\n" ); document.write( " $\"x+%2B+x+%2B+x+%2B+log%28a%2C+%285%29%29+-+3\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"3x+%2B+log%28a%2C+%285%29%29+-+3\"$
\n" ); document.write( "All we have left to do is to express $\"log%28a%2C+%285%29%29\"$ in terms of x and/or y. Since 2 is not a factor of 5 it would seem that x will not help us with this. 25 is not a factor of 5 either so at first glance it would seem that y will not help either. But there is a connection between 5 and 25. $\"5%5E2+=+25\"$ and $\"sqrt%2825%29+=+5\"$ . We can use this connection to take the equation for y and modify it to give us an expression for $\"log%28a%2C+%285%29%29\"$ :
\n" ); document.write( " $\"y+=+log%28a%2C+%2825%29%29\"$
\n" ); document.write( "Square roots are exponents of 1/2. So somehow we need to get an exponent of 1/2 on the 25. The only legitimate way is a little tricky. We cannot just raise both sides to the 1/2 power. This would put the exponent on the logarithm, not on the 25 in the argument of the logarithm. But we have yet another property of logarithms, $\"q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29\"$ , which comes to our rescue. This lets us move a coefficient in front of the logarithm into the argument as an exponent! So if we can introduce a coefficient of 1/2 we can then use this property to move it into the argument as the exponent of 25.

\n" ); document.write( "To introduce the coefficient of 1/2 all we need to do is multiply both sides of our equation by 1/2:
\n" ); document.write( " $\"%281%2F2%29y+=+%281%2F2%29log%28a%2C+%2825%29%29\"$
\n" ); document.write( "Now we can use the property:
\n" ); document.write( " $\"%281%2F2%29y+=+log%28a%2C+%2825%5E%281%2F2%29%29%29\"$
\n" ); document.write( "and simplify:
\n" ); document.write( " $\"%281%2F2%29y+=+log%28a%2C+%285%29%29\"$
\n" ); document.write( "Now we can return to our expression:
\n" ); document.write( " $\"3x+%2B+log%28a%2C+%285%29%29+-+3\"$
\n" ); document.write( "and substitute for $\"log%28a%2C+%285%29%29\"$ :
\n" ); document.write( " $\"3x+%2B+%281%2F2%29y+-+3\"$ \n" ); document.write( "

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