document.write( "Question 240807: log 8 (sqrt 8^9) = n \n" ); document.write( "

Algebra.Com's Answer #176414 by jsmallt9(3759) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"log%288%2C+%28sqrt%288%5E9%29%29%29+=+n\"$
\n" ); document.write( "As usual, there are several ways to solve this. Maybe the quickest way is based on

knowing our exponents (including fractional exponents) well
seeing that the argument of the base 8 logarithm is a power of 8
knowing that you do not need a calculator to find the base 8 logarithm of a power of 8

\n" ); document.write( "Since square roots are the same as raising to the 1/2 power we can rewrite the argument:
\n" ); document.write( " $\"log%288%2C+%28%288%5E9%29%5E%281%2F2%29%29%29+=+n\"$
\n" ); document.write( "Using the property of exponents, $\"%28a%5Ep%29%5Eq+=+a%5E%28p%2Aq%29\"$ , we get:
\n" ); document.write( " $\"log%288%2C+%288%5E%289%2F2%29%29%29+=+n\"$
\n" ); document.write( "Using the property of logarithms, $\"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\"$ , we can move the exponent out in front:
\n" ); document.write( " $\"%289%2F2%29log%288%2C+%288%29%29+=+n\"$
\n" ); document.write( "Since by definition $\"log%28a%2C+%28a%29%29+=+1\"$ :
\n" ); document.write( " $\"9%2F2+=+n\"$
\n" ); document.write( " \n" ); document.write( "

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