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4^x + 4^-x = 5/2. What is x
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\n" ); document.write( "Here is one approach: 4^-x is the reciprocal of 4^x
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\n" ); document.write( "We can present the problem as \"What number + it's reciprocal = 5/2?\"
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\n" ); document.write( "Let a = 4^x and write the problem:
\n" ); document.write( "a + 1/a = 5/2
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\n" ); document.write( "a^2 + 1 = 5a/2; multiplied equation by a
\n" ); document.write( "2a^2 + 2 = 5a; multiplied equation by 2
\n" ); document.write( "2a^2 - 5a + 2 = 0; a quadratic equation
\n" ); document.write( "(2a - 1)(a - 2) = 0; factored
\n" ); document.write( "2a = +1
\n" ); document.write( "a = 1/2
\n" ); document.write( "and
\n" ); document.write( "a = +2
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\n" ); document.write( "use the last solution: 4^x = a
\n" ); document.write( "4^x = 2
\n" ); document.write( "We know 2 is the sqrt of 4 therefore
\n" ); document.write( "x = 1/2
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\n" ); document.write( "Check:
\n" ); document.write( "4^1/2 + 4^(-1/2) =
\n" ); document.write( "2 + (1/2) = 5/2
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\n" ); document.write( "You can check and see if the other solution a = 1/2 will work.
\n" ); document.write( "4^x = (1/2)
\n" ); document.write( "Use natural logs for this:
\n" ); document.write( "x*ln(4) = ln(1/2)
\n" ); document.write( "1.38629x = -.693147
\n" ); document.write( "x = -.693147/1.38629
\n" ); document.write( "x = -.500000 or -1/2
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\n" ); document.write( "That makes the problem 4^(-1/2) + 4^-(-1/2) = 5/2
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\n" ); document.write( "Hope this round about way was some help to you. \n" ); document.write( "