document.write( "Question 238662: (log4(log4^x))=-4 \n" ); document.write( "

Algebra.Com's Answer #175424 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"log%284%2C+%28log%284%2C+%28x%29%29%29%29+=+-+4\"$
\n" ); document.write( "Solving equations of the form
\n" ); document.write( "log(some-expression-with-a-variable) = another-expression
\n" ); document.write( "is usually done by rewriting the equaiton in exponential form. To rewrite logarithmic equations in logarithmic form we need to remember that $\"log%28a%2C+%28p%29%29+=+q\"$ is equivalent to $\"p+=+a%5Eq\"$

\n" ); document.write( "Since your equation has a logarithm within a logarithm, we will have to do this twice. Rewriting the outer logarithm in exponential form we get:
\n" ); document.write( " $\"log%284%2C+%28x%29%29+=+4%5E%28-4%29\"$
\n" ); document.write( "Since $\"4%5E%28-4%29+=+1%2F%284%5E4%29\"$ and $\"4%5E4+=+4%2A4%2A4%2A4+=+256\"$ , the right side simplifies to:
\n" ); document.write( " $\"log%284%2C+%28x%29%29+=+1%2F256\"$
\n" ); document.write( "Now we will rewrite the remaining logarithm in exponential form:
\n" ); document.write( " $\"x+=+4%5E%281%2F256%29\"$
\n" ); document.write( "This may be an acceptable form for the answer. But it can be simplified a little. First we can \"reduce\" the exponent using a bit of cleverness and a good understanding of fractional exponents:
\n" ); document.write( "Factor the exponent:
\n" ); document.write( " $\"x+=+4%5E%28%281%2F2%29%281%2F128%29%29\"$
\n" ); document.write( "Use the property $\"a%5E%28p%2Aq%29+=+%28a%5Ep%29%5Eq\"$ to rewrite the abvoe as a power of a power:
\n" ); document.write( " $\"x+=+%284%5E%281%2F2%29%29%5E%281%2F128%29\"$
\n" ); document.write( "Since $\"4%5E%281%2F2%29+=+sqrt%284%29\"$ and $\"sqrt%284%29+=+2\"$ , we can replace $\"4%5E%281%2F2%29\"$ with 2. (This is why we factored 1/2 out of the exponent.)
\n" ); document.write( " $\"x+=+2%5E%281%2F128%29\"$
\n" ); document.write( "This is a simplified form of the answer. With a fractional exponent, we could write this in radical form:
\n" ); document.write( " $\"x+=+root%28128%2C+2%29\"$ \r
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