document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=172999>Question 172999</A>: <I><SPAN STYLE=\"word-break: break-all;\"> For the exponential function e^x and logarithmic function log x, graphically show the effect if x is doubled, can someone help me with the problem, I do not understand what they are asking.  Thank you!!! \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #127869 by </B><A HREF=/tutors/aboutme.mpl?userid=gonzo><B>gonzo(654)</B></A><img src=\"/images/stars/stars-11.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=gonzo><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=127869\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> they want to know the impact of doubling x.<BR>\n" );
document.write( "i take this to mean:<BR>\n" );
document.write( "if you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you double x, then you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x%29 ALIGN=MIDDLE  ALT=\"e%5E%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "likewise:<BR>\n" );
document.write( "if you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cx%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and you double x, then you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C2x%29 ALIGN=MIDDLE  ALT=\"log%2810%2C2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "first i will graph <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x%29 ALIGN=MIDDLE  ALT=\"e%5E%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to see what the impact would be:<BR>\n" );
document.write( "look below the graph for further comments.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28800%2C800%2C-3%2C5%2C-2%2C20%2C2.718281828%5Ex%2C2.718281828%5E%282x%29%29 ALIGN=MIDDLE  ALT=\"graph%28800%2C800%2C-3%2C5%2C-2%2C20%2C2.718281828%5Ex%2C2.718281828%5E%282x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "the faster rising curve is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x%29 ALIGN=MIDDLE  ALT=\"e%5E%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "when x = 0, the values of both curves is the same.<BR>\n" );
document.write( "when x = 1, the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 2.718... and the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x%29 ALIGN=MIDDLE  ALT=\"e%5E%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 7.389...<BR>\n" );
document.write( "when x = 2, the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 7.389... and the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x%29 ALIGN=MIDDLE  ALT=\"e%5E%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is 54.598...<BR>\n" );
document.write( "the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E2x ALIGN=MIDDLE  ALT=\"e%5E2x\"   DISABLED_style= \"max-width:90%; height:auto;\" > appears to be the value of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28e%5Ex%29%5E2 ALIGN=MIDDLE  ALT=\"%28e%5Ex%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is should since<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28a%5Eb%29%5Ec ALIGN=MIDDLE  ALT=\"%28a%5Eb%29%5Ec\"   DISABLED_style= \"max-width:90%; height:auto;\" > equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E%28b%2Ac%29 ALIGN=MIDDLE  ALT=\"a%5E%28b%2Ac%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > by the laws of exponentiation.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "when x = -.5, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > = .6065 and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x%29 ALIGN=MIDDLE  ALT=\"e%5E%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = .3678.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=.6065%5E2 ALIGN=MIDDLE  ALT=\".6065%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = .3678 so the ratio holds even though <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%282x%29 ALIGN=MIDDLE  ALT=\"e%5E%282x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is smaller than <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > at this point.<BR>\n" );
document.write( "---<BR>\n" );
document.write( "you can see this on the graph.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Cx%29 ALIGN=MIDDLE  ALT=\"log%2810%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C2x%29 ALIGN=MIDDLE  ALT=\"log%2810%2C2x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is shown now.<BR>\n" );
document.write( "look below the graph for further comments.<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28800%2C800%2C-2%2C40%2C-1%2C2%2Clog%2810%2Cx%29%2Clog%2810%2C2x%29%29 ALIGN=MIDDLE  ALT=\"graph%28800%2C800%2C-2%2C40%2C-1%2C2%2Clog%2810%2Cx%29%2Clog%2810%2C2x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "this graph doesn't show such a dramatic change in the value of log(x) versus log(2x)<BR>\n" );
document.write( "example:<BR>\n" );
document.write( "when x = 10, log(x) = 1, and log(2x) = 1.301...<BR>\n" );
document.write( "when x = 100, log(x) = 2, and log(2x) = 2.301...<BR>\n" );
document.write( "when x = 1000, log(x) = 3, and log(2x) = 3.301...<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "this is because when x is an exponent, it has a very large impact on the result.<BR>\n" );
document.write( "when x is the result of exponentiation, it has a very small impact on the result.<BR>\n" );
document.write( "when x is the exponent, this is what happens:<BR>\n" );
document.write( "if you take x = 1, then 10^1 = 10<BR>\n" );
document.write( "if you take x = 10, then 10^10 = 10000000000<BR>\n" );
document.write( "there's a big difference.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "when x is the result of exponentiation, this is what happens:<BR>\n" );
document.write( "if you take x = 1000, then 10^y = 1000 which means y = 3.<BR>\n" );
document.write( "if you double x to be 2000, then 10^y = 2000 which means y = 3.301029996<BR>\n" );
document.write( "doubling the answers causes a very small increase in the exponent.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "y = 10^x means x is the exponent<BR>\n" );
document.write( "y = log(x) means x is the answer.<BR>\n" );
document.write( "remember:<BR>\n" );
document.write( "y = log(x) if and only if x = 10^y<BR>\n" );
document.write( "x is the result of exponentiation when you say y = log(x)<BR>\n" );
document.write( "x is the exponent itself when you say y = 10^x.<BR>\n" );
document.write( "----<BR>\n" );
document.write( "best i can do.<BR>\n" );
document.write( "hope it makes sense.<BR>\n" );
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