document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=433406>Question 433406</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A = ln(x)-Bx<BR>\n" );
document.write( "Express x in terms of A and B.<BR>\n" );
document.write( "Please help.... thanks in advance..... </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #300523 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=300523\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A+=+ln%28x%29+-+ln%28e%5E%28Bx%29%29+=+ln%28x%2F%28e%5E%28Bx%29%29%29 ALIGN=MIDDLE  ALT=\"A+=+ln%28x%29+-+ln%28e%5E%28Bx%29%29+=+ln%28x%2F%28e%5E%28Bx%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5EA+=+x%2Fe%5E%28Bx%29 ALIGN=MIDDLE  ALT=\"e%5EA+=+x%2Fe%5E%28Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5EA+=+xe%5E%28-Bx%29 ALIGN=MIDDLE  ALT=\"e%5EA+=+xe%5E%28-Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Multiply both sides by -B\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-Be%5EA+=+-Bxe%5E%28-Bx%29 ALIGN=MIDDLE  ALT=\"-Be%5EA+=+-Bxe%5E%28-Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "There is a certain function called the Lambert W-function defined as the inverse of the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28z%29+=+ze%5Ez ALIGN=MIDDLE  ALT=\"f%28z%29+=+ze%5Ez\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Here, we have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28-Bx%29+=+-Bxe%5E%28-Bx%29 ALIGN=MIDDLE  ALT=\"f%28-Bx%29+=+-Bxe%5E%28-Bx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > so it follows that if we take the W-function of both sides,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-Bx+=+W%28-Be%5EA%29 ALIGN=MIDDLE  ALT=\"-Bx+=+W%28-Be%5EA%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2FB%29W%28-Be%5EA%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2FB%29W%28-Be%5EA%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This is pretty much the simplest form that one can express x in, even though the W-function is somewhat obscure. There is, of course, using Newton's method, but it is very unlikely that a real number will be obtained.\r<BR>\n" );
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document.write( "Also, I checked my answer on Wolfram Alpha by putting arbitrary values for A and B (e.g. A = 253, B = 19), and it came out with the solution <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28-1%2F19%29W%28-19e%5E253%29 ALIGN=MIDDLE  ALT=\"x+=+%28-1%2F19%29W%28-19e%5E253%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "http://www.wolframalpha.com/input/?i=253+%3D+ln%28x%29+-+19x\r<BR>\n" );
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document.write( "Wikipedia article about the W-function:<BR>\n" );
document.write( "http://en.wikipedia.org/wiki/Lambert_W_function\r<BR>\n" );
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