document.write( "Question 824341: Given that log (base 7)x^2 = p and log (base 7) xy^2 = q, find log (base 7) (cube root over xy) in terms of p and q.
\n" ); document.write( "hope you'll get my question right! \n" ); document.write( "

Algebra.Com's Answer #496332 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"log%287%2C+%28x%5E2%29%29+=+p\"$ and $\"log%287%2C+%28xy%5E2%29%29+=+q\"$
\n" ); document.write( "What we are going to do is...

Solve the above equations to get expressions for $\"log%287%2C+%28x%29%29\"$ and $\"log%287%2C+%28y%29%29\"$ .
Rewrite $\"log%287%2C+%28root%283%2C+xy%29%29%29\"$ in terms of $\"log%287%2C+%28x%29%29\"$ and $\"log%287%2C+%28y%29%29\"$ .
Substitute in the expressions for $\"log%287%2C+%28x%29%29\"$ and $\"log%287%2C+%28y%29%29\"$ from step 1 into the expression we got in step 2.
Simplify.

Solving $\"log%287%2C+%28x%5E2%29%29+=+p\"$ for $\"log%287%2C+%28x%29%29\"$ :
\n" ); document.write( "Using a property of logarithms, $\"log%28a%2C+%28j%5En%29%29+=+n%2Alog%28a%2C+%28j%29%29\"$ , to move the exponent out in front:
\n" ); document.write( " $\"2log%287%2C+%28x%29%29+=+p\"$
\n" ); document.write( "Dividing both sides by 2:
\n" ); document.write( " $\"log%287%2C+%28x%29%29+=+p%2F2\"$

\n" ); document.write( "Solving $\"log%287%2C+%28xy%5E2%29%29+=+q\"$ for $\"log%287%2C+%28y%29%29\"$ :
\n" ); document.write( "Using another property of logarithms, $\"log%28a%2C+%28j%2Ak%29%29+=+log%28a%2C+%28j%29%29+%2B+log%28a%2C+%28k%29%29\"$ , we can split the argument into separate logs:
\n" ); document.write( " $\"log%287%2C+%28x%29%29+%2B+log%287%2C+%28y%5E2%29%29+=+q\"$
\n" ); document.write( "Substituting in the expression we got for $\"log%287%2C+%28x%29%29\"$ :
\n" ); document.write( " $\"p%2F2+%2B+log%287%2C+%28y%5E2%29%29+=+q\"$
\n" ); document.write( "Using the property to move the exponent:
\n" ); document.write( " $\"p%2F2+%2B+2log%287%2C+%28y%29%29+=+q\"$
\n" ); document.write( "Multiplying each side by 1/2:
\n" ); document.write( " $\"p%2F4+%2B+log%287%2C+%28y%29%29+=+q%2F2\"$
\n" ); document.write( "Subtracting p/4:
\n" ); document.write( " $\"log%287%2C+%28y%29%29+=+q%2F2-p%2F4\"$

\n" ); document.write( "Rewriting $\"log%287%2C+%28root%283%2C+xy%29%29%29\"$ in terms of $\"log%287%2C+%28x%29%29\"$ and $\"log%287%2C+%28y%29%29\"$ :
\n" ); document.write( "Since a cube root is the same as an exponent of 1/3:
\n" ); document.write( " $\"log%287%2C+%28%28xy%29%5E%281%2F3%29%29%29\"$
\n" ); document.write( "Moving the exponent:
\n" ); document.write( " $\"%281%2F3%29log%287%2C+%28xy%29%29\"$
\n" ); document.write( "Split the argument:
\n" ); document.write( " $\"%281%2F3%29%28log%287%2C+%28x%29%29+%2B+log%287%2C+%28y%29%29%29\"$
\n" ); document.write( "Substituting in our expressions for the two logs:
\n" ); document.write( " $\"%281%2F3%29%28p%2F2+%2B+q%2F2+-+p%2F4%29\"$
\n" ); document.write( "Simplifying...
\n" ); document.write( " $\"%281%2F3%29%282p%2F4+%2B+q%2F2+-+p%2F4%29\"$
\n" ); document.write( " $\"%281%2F3%29%28p%2F4+%2B+q%2F2%29\"$
\n" ); document.write( " $\"p%2F12+%2B+q%2F6%29\"$ \n" ); document.write( "

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