document.write( "Question 557237: find the inverse: y=6+log(base 2)^x \n" ); document.write( "
Algebra.Com's Answer #362511 by Theo(13342)\"\" \"About 
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your equation is:
\n" ); document.write( "y = 6 + log(2,x^2)
\n" ); document.write( "subtract 6 from both sides of the equation to get:
\n" ); document.write( "log(2,x^2) = (y-6)
\n" ); document.write( "this equation is true if and only if:
\n" ); document.write( "2^(y-6) = x^2
\n" ); document.write( "replace x with y and y with x to get:
\n" ); document.write( "2^(x-6) = y^2
\n" ); document.write( "take the square root of both sides of this equation to get:
\n" ); document.write( "y = +/- sqrt(2^(x-6)
\n" ); document.write( "your 2 equations that should be inverses of each other are:
\n" ); document.write( "y = 6 + log(2,x^2)
\n" ); document.write( "y = +/- sqrt(2^(x-6)
\n" ); document.write( "graph of the first equation looks like this:
\n" ); document.write( "\"graph%28600%2C600%2C-15%2C15%2C-15%2C15%2C6+%2B+log%282%2Cx%5E2%29%29\"
\n" ); document.write( "graph of the second equation looks like this:
\n" ); document.write( "
\n" ); document.write( "graph of these 2 equations superimposed on each other along with the graph of the equation y = x looks like this:
\n" ); document.write( "
\n" ); document.write( "they are inverse equation if the point (x,y) in one equation equals the point (y,x) in the second equation.
\n" ); document.write( "we'll choose the value of x in the first equation to be equal to 6
\n" ); document.write( "when x = 6, the value of the first equation of:
\n" ); document.write( "y = 6 + log(2,x^2) becomes:
\n" ); document.write( "y = 6 + log(2,6^2) which becomes:
\n" ); document.write( "y = 6 + log(2,36) which becomes:
\n" ); document.write( "y = 11.169925001
\n" ); document.write( "the solution for the first equation is (x,y) = 6,11.169925001)
\n" ); document.write( "to see if the second equation is truly an inverse of the first equation, we solve the second equation for x equal to the value of y which means that:
\n" ); document.write( "x = 11.169925001.
\n" ); document.write( "when x = 5.169925001, the second equation of:
\n" ); document.write( "y = +/- sqrt(2^(x-6) becomes:
\n" ); document.write( "y = +/- sqrt(2^(5.169925001-6) which becomes:
\n" ); document.write( "y = +/- sqrt(2^(-.8300749986) which becomes:
\n" ); document.write( "y = +/- 6
\n" ); document.write( "when x = -6 in the first equation of:
\n" ); document.write( "y = 6 + log(2,x^2) becomes:
\n" ); document.write( "y = 6 + log(2,36) which, once again, give you a value of:
\n" ); document.write( "y = 11.169925001.
\n" ); document.write( "we get the following:
\n" ); document.write( "when x = 6 in the first equation, y = 11.169925001.
\n" ); document.write( "when y = 11.169925001 in the second equation, y = 6
\n" ); document.write( "we get f(x,y) = f(y,x)
\n" ); document.write( "when x = -6 in the first equation, y = 11.169925001.
\n" ); document.write( "when y = 11.169925001 in the second equation, y = -6
\n" ); document.write( "we get f(x,y)( = f(y,x)
\n" ); document.write( "note that the inverse equation is not a function because there are more than one value of y for each value of x.
\n" ); document.write( "the original equation is a function, but the inverse equation is a relation.\r
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