document.write( "Question 975026: Help solve this equation
\n" ); document.write( "4^c=5(2^(c-1))+6 \n" ); document.write( "

Algebra.Com's Answer #597008 by KMST(5347) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"f%28c%29=4%5Ec=%282%5E2%29%5Ec=2%5E2%5E%282c%29\"$ is an exponential function,
\n" ); document.write( "with a shape like $\"graph%28300%2C300%2C-.2%2C.8%2C-.1%2C.9%2C0.1%2A2%5E%285x%29%29\"$ , always increasing.
\n" ); document.write( "So is $\"g%28c%29=5%282%5E%28c-1%29%29%2B6\"$ , but $\"g%28c%29\"$ does not increase as sharply,
\n" ); document.write( "because the exponent is half as much.
\n" ); document.write( "For $\"c=0\"$ , $\"g%280%29=5%282%5E%28-1%29%29%2B6=5%281%2F2%29%2B6=5%2F2%2B6%3Eh%280%29=4%5E0=1\"$ ,
\n" ); document.write( "but as $\"c\"$ increases, at some value of $\"c%3E0\"$ ,
\n" ); document.write( " $\"f%28c%29\"$ eventually catches up with $\"g%28c%29\"$ , and then surpasses it.
\n" ); document.write( "So there is one, and just one value of $\"c\"$ where $\"f%28c%29=4%5Ec=5%282%5E%28c-1%29%29%2B6=g%28c%29\"$ ,
\n" ); document.write( "for lesser values of $\"c\"$ , $\"f%28c%29%3Cg%28c%29\"$ ,
\n" ); document.write( "and for greater values of $\"c\"$ , $\"f%28c%29%3Eg%28c%29\"$ .
\n" ); document.write( "
\n" ); document.write( "Where is $\"f%28c%29=g%28c%29\"$ <---> $\"4%5Ec=5%282%5E%28c-1%29%29%2B6\"$ ?
\n" ); document.write( "It is for some $\"c%3E0\"$ , but for what value of $\"c%3E0\"$ ?
\n" ); document.write( "How are you expected to find Out?
\n" ); document.write( "Guess and check? Graphic calculator?
\n" ); document.write( "
\n" ); document.write( " $\"4%5Ec=5%282%5E%28c-1%29%29%2B6\"$ <---> $\"2%5E%282c%29=5%282%5E%28c-1%29%29%2B6\"$ <---> $\"2%5E%282c%29%2F2=%285%282%5E%28c-1%29%29%2B6%29%2F2\"$ <---> $\"2%5E%282c-1%29=5%282%5E%28c-1%29%29%2F2%2B6%2F2\"$ <---> $\"2%5E%282c-1%29=5%282%5E%28c-2%29%29%2B3\"$
\n" ); document.write( "If we only consider positive integer values of $\"c\"$ ,
\n" ); document.write( "We realize that $\"2%5E%282c-1%29\"$ is always an even number.
\n" ); document.write( "Also, as long as $\"c-2%3E=1\"$ <--> $\"c%3E=3\"$ , $\"5%282%5E%28c-2%29%29\"$ is even and $\"5%282%5E%28c-2%29%29%2B3\"$ is odd,
\n" ); document.write( "so we cannot get $\"4%5Ec=5%282%5E%28c-1%29%29%2B6\"$ <---> $\"2%5E%282c-1%29=5%282%5E%28c-2%29%29%2B3\"$ with integers such that $\"c%3E=3\"$ .
\n" ); document.write( "For $\"5%282%5E%28c-2%29%29%2B3\"$ to be even, $\"5%282%5E%28c-2%29%29\"$ must be odd,
\n" ); document.write( "and that only happens when $\"c=2\"$ <--> $\"c-2=0\"$ <--> $\"2%5E%28c-2%29=1\"$ <--> $\"5%282%5E%28c-2%29%29=5\"$ <--> $\"5%282%5E%28c-2%29%29%2B3=5%2B3=8\"$ .
\n" ); document.write( "For that value of $\"c\"$ , $\"c=2\"$ , $\"2%5E%282c-1%29=2%5E%282%2A2-1%29=2%5E%284-1%29=2%5E3=8\"$ ,
\n" ); document.write( "and then $\"2%5E%282c-1%29=5%282%5E%28c-2%29%29%2B3\"$ <---> $\"4%5Ec=5%282%5E%28c-1%29%29%2B6\"$ .
\n" ); document.write( "Since $\"c=2\"$ is the only integer solution of $\"4%5Ec=5%282%5E%28c-1%29%29%2B6\"$ <---> $\"2%5E%282c-1%29=5%282%5E%28c-2%29%29%2B3\"$ ,
\n" ); document.write( "and because we knew that $\"4%5Ec=5%282%5E%28c-1%29%29%2B6\"$ had one and just one solution,
\n" ); document.write( " $\"highlight%28c=2%29\"$ is THE solution to $\"4%5Ec=5%282%5E%28c-1%29%29%2B6\"$ . \n" ); document.write( "

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