document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=417286>Question 417286</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a^x = 1 + bx\r<BR>\n" );
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document.write( "a and b are known; find x\r<BR>\n" );
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document.write( "x=0 is a trivial solution that does not depend on a or b, so this doesn't help.\r<BR>\n" );
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document.write( "I already have a logarithmic expansion to a solvable cubic equation; and I have a heuristic. If there is nothing more elegant, your conclusion to that effect would be informative.\r<BR>\n" );
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document.write( "Thanks. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #292153 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=292153\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> You're given an exponential function (which is transcendental) and an algebraic function in the same equation. Transcendental functions do not combine well with algebraic functions. The best way (other than using a calculator) is to use Newton's method.\r<BR>\n" );
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document.write( "Newton's method defines a recursive sequence <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5Bi%5D ALIGN=MIDDLE  ALT=\"x%5Bi%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" > with an initial \"guess\" <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B0%5D ALIGN=MIDDLE  ALT=\"x%5B0%5D\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Newton's method says that, to find the zeros of a function f(x) = 0 with a given <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5B0%5D ALIGN=MIDDLE  ALT=\"x%5B0%5D\"   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%5Bi%2B1%5D+=+x%5Bi%5D+-+%28f%28x%5Bi%5D%29%2Ff%5E%281%29+%28x%5Bi%5D%29%29 ALIGN=MIDDLE  ALT=\"x%5Bi%2B1%5D+=+x%5Bi%5D+-+%28f%28x%5Bi%5D%29%2Ff%5E%281%29+%28x%5Bi%5D%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Here, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E%281%29+%28x%5Bi%5D%29 ALIGN=MIDDLE  ALT=\"f%5E%281%29+%28x%5Bi%5D%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > represents f'(x[i]), the derivative of the function <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29+=+a%5Ex+-+bx+-+1 ALIGN=MIDDLE  ALT=\"f%28x%29+=+a%5Ex+-+bx+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Here, the derivative of f(x) is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%5E%281%29%28x%29+=%28ln%28a%29%29a%5Ex+-+b ALIGN=MIDDLE  ALT=\"f%5E%281%29%28x%29+=%28ln%28a%29%29a%5Ex+-+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >. </SPAN>\n" );
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