document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1074623>Question 1074623</A>: <I><SPAN STYLE=\"word-break: break-all;\"> given: f(x)=(1/4)^x. how do I write down the equation of f^-1(x) in the form y=... </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #689275 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=689275\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> start with f(x) = (1/4)^x.\r<BR>\n" );
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document.write( "set y = f(x) and the equation becomes y = (1/4)^x.\r<BR>\n" );
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document.write( "replace y with x and x with y and the equation becomes x = (1/4)^y.\r<BR>\n" );
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document.write( "now you want to solve for y.\r<BR>\n" );
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document.write( "x = (1/4)^y if and only if log<sub>(1/4)</sub>(x) = y\r<BR>\n" );
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document.write( "your equation becomes y = log<sub>(1/4)</sub>(x)\r<BR>\n" );
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document.write( "set y = f<sup>-1</sup>(x) and the equation becomes f<sup>-1</sup>(x) = log<sub>(1/4)</sub>(x)\r<BR>\n" );
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document.write( "the graph of both equations is shown below.<BR>\n" );
document.write( "since they are inverse equations, they are reflections about the equation of y = x.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2017/032803.jpg\" alt=\"$$$\" </>\r<BR>\n" );
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document.write( "they are inverse equations if (x,y) in f(x) is equal to (y,x) in f<sup>-1</sup>(x).\r<BR>\n" );
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document.write( "the intersection of the both graphs with a line perpendicular to the line y = x shows this to be true.\r<BR>\n" );
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document.write( "the red curve is the normal equation and the blue curve is the inverse equation.\r<BR>\n" );
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document.write( "the reflection of the point (-1.448,7.448) in the normal equation is the point (7.448,-1.448) in the inverse equation, confirming that they are inverse equations.\r<BR>\n" );
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document.write( "in general, if you start with y = b<sup>x</sup>, and you want to find the inverse equation, then replace x with y and y with x to get the equation x = b<sup>y</sup>\r<BR>\n" );
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document.write( "this is true if and only if y = log<sub>b</sub>(x)\r<BR>\n" );
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document.write( "the inverse equation to y = b<sup>x</sup> is therefore y = log<sub>b</sub>(x).\r<BR>\n" );
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document.write( "here's a nice reference on logarithmix and exponential functions you might find useful.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.themathpage.com/aprecalc/logarithmic-exponential-functions.htm\" target = \"_blank\">http://www.themathpage.com/aprecalc/logarithmic-exponential-functions.htm</a>\r<BR>\n" );
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