document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1118206>Question 1118206</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is the value of x in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2Cx%29=log%284%2Cx%5E2%29 ALIGN=MIDDLE  ALT=\"log%282%2Cx%29=log%284%2Cx%5E2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #733464 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=733464\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> somehow these wind up being identical when x is positive.\r<BR>\n" );
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document.write( "the difference is that log4(x^2) allows the value of x to be positive or negative, while log2(x) allows the value of x to be positive only.\r<BR>\n" );
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document.write( "so, if you restrict your domain to positive values of x, the 2 equations are identical.\r<BR>\n" );
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document.write( "since they are identical, then x can be any positive value.\r<BR>\n" );
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document.write( "our equation is log2(x) = log4(x^2).\r<BR>\n" );
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document.write( "it's clear that x has to be positive, since otherwise log2(x) would not provide a real answer, since x has to be > 0.\r<BR>\n" );
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document.write( "log2(x) = a if and only if x^a = x.\r<BR>\n" );
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document.write( "that's from the basic definition of of what a log is.\r<BR>\n" );
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document.write( "likewise, log4(x^2) = b if and only if 4^b = x^2.\r<BR>\n" );
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document.write( "4 is equal to 2^2, therefore 4^b is equal to (2^2)^b and you get:\r<BR>\n" );
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document.write( "(2^2)^b = x^2\r<BR>\n" );
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document.write( "(2^2)^b is equal to 2^(2b) which is equal to (2^b)^2.\r<BR>\n" );
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document.write( "you get (2^b)^2 = x^2.\r<BR>\n" );
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document.write( "take the square root of both sides of the equaiton to get:\r<BR>\n" );
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document.write( "2^b = plus or minus x.\r<BR>\n" );
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document.write( "when 2^b = plus x, the basic definition of logs states that 2^b = x if and only if b = log2(x).\r<BR>\n" );
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document.write( "that's ok, since x is positive.\r<BR>\n" );
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document.write( "however, 2^b = -x leads to log2(-x) = b which can't be, since x has to be positive.\r<BR>\n" );
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document.write( "therefore, when x is posiive, you get:\r<BR>\n" );
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document.write( "2^a = x and you get 2^b = x.\r<BR>\n" );
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document.write( "this means that a must be equal to b.\r<BR>\n" );
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document.write( "the two equations become identical as long as x is positive.\r<BR>\n" );
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document.write( "so, you get:\r<BR>\n" );
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document.write( "log2(x) = a if and only if 2^a = x.\r<BR>\n" );
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document.write( "square both sides of that equation and you get (2^a)^2 = x^2\r<BR>\n" );
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document.write( "that's the same as 2^2a = x^2 which is the same as (2^2)^a = x^2 which is the same as 4^a = x^2.\r<BR>\n" );
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document.write( "the equations are identical as long as x is positive, therefore x can be any value as long as it is positive.\r<BR>\n" );
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document.write( "i'm not sure if i explained it well, but that's what i'm seeing that the answer is.\r<BR>\n" );
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document.write( "this can be seen in the following 3 graphs.\r<BR>\n" );
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document.write( "the first graph is y = log2(x).\r<BR>\n" );
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document.write( "the second graph is y = log4(x^2).\r<BR>\n" );
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document.write( "the third graph is both equations shown on the same graph.\r<BR>\n" );
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document.write( "you can see that the black line of y = log2(x) has turned red after it was superimposed  on by the orange line of y = log4(x^2).\r<BR>\n" );
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document.write( "that only happens on the right side of the graph because y = log2(x) is only valid when x is greater than 0.\r<BR>\n" );
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document.write( "y = log4(x^2) is valid for all real values of x except, i think, when x = 0.\r<BR>\n" );
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document.write( "neither graph is valid when x = 0.\r<BR>\n" );
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document.write( "here's the graphs.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/060401.jpg\" alt=\"$$$\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2018/060403.jpg\" alt=\"$$$\" >\r<BR>\n" );
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