document.write( "Question 1091577: log1/2 16 = -4 \n" ); document.write( "

Algebra.Com's Answer #705997 by KMST(5328) $\"\"$ $\"About$

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If you were not given the answer to $\"log%281%2F2%2C16%29\"$ $\"%22=%22\"$ $\"%22%3F%22\"$ , there is a way to figure out that the answer is $\"4\"$ .
\n" ); document.write( "If you are given that statement $\"log%281%2F2%2C16%29\"$ $\"%22=%22\"$ $\"-4\"$ , you can verify/prove that is true by showing that $\"%281%2F2%29%5E%28-4%29\"$ $\"%22=%22\"$ $\"16\"$ .
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\n" ); document.write( "FINDING THE ANSWER:
\n" ); document.write( "If you need to calculate logarithms with an cumbersome base $\"c\"$ ,
\n" ); document.write( "you can convert them to a more helpful base $\"b\"$ , by using the fact that
\n" ); document.write( " $\"log%28c%2Cx%29\"$ $\"%22=%22\"$ $\"log%28b%2Cx%29%2Flog%28b%2Cc%29\"$ .
\n" ); document.write( "That is a theorem that can be formally proven,
\n" ); document.write( "and as a formula is useful, and easy to remember.
\n" ); document.write( "Applied to this case, using $\"2\"$ as your helpful base
\n" ); document.write( " $\"log%28%221+%2F+2%22%2C16%29\"$ $\"%22=%22\"$ $\"log%282%2C16%29%2Flog%282%2C%221+%2F+2%22%29\"$ $\"%22=%22\"$ $\"4%2F%28-1%29\"$ $\"%22=%22\"$ $\"highlight%28-4%29\"$ .
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\n" ); document.write( "PROVING GIVEN STATEMENT:
\n" ); document.write( "Here is one way to prove that $\"%281%2F2%29%5E%28-4%29\"$ $\"%22=%22\"$ $\"16\"$ :
\n" ); document.write( "The fraction $\"1%2F2\"$ is $\"1\"$ divided by $\"2\"$ ,
\n" ); document.write( "and you know that with exponents,
\n" ); document.write( " $\"%28a%2Fb%29%5En=a%5En%2Fb%5En\"$ is true, as long as $\"b%3C%3E0\"$ .
\n" ); document.write( "So, in this case,
\n" ); document.write( " $\"%281%2F2%29%5E%28-4%29\"$ $\"%22=%22\"$ $\"1%5E%28-4%29%2F2%5E%28-4%29\"$ $\"%22=%22\"$ $\"1%2F2%5E%28-4%29\"$ ,
\n" ); document.write( "and you know that by the definition of negative exponents
\n" ); document.write( " $\"2%5E%28-4%29\"$ $\"%22=%22\"$ $\"1%2F2%5E4\"$ $\"%22=%22\"$ $\"1%2F16\"$ ,
\n" ); document.write( "so $\"%281%2F2%29%5E%28-4%29\"$ $\"%22=%22\"$ $\"1%2F%221+%2F+16%22\"$ $\"%22=%22\"$ $\"16\"$ .
\n" ); document.write( "You know that
\n" ); document.write( " $\"%281%2F2%29%5E4=%281%2F2%29%2A%281%2F2%29%2A%281%2F2%29%2A%281%2F2%29=1%2F16\"$ ,
\n" ); document.write( "and you should know that
\n" ); document.write( " $\"1%2F%221+%2F+16%22=16\"$
\n" ); document.write( "
\n" ); document.write( "If you can think of $\"1%2F2\"$ only as a fraction,
\n" ); document.write( "you still would not that the way negative exponents are defined,
\n" ); document.write( " $\"%281%2F2%29%5E%28-4%29\"$ $\"%22=%22\"$ $\"1%2F%22%28+1+%2F+2+%29%22%5E4\"$ \n" ); document.write( "

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