document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=212950>Question 212950</A>: <I><SPAN STYLE=\"word-break: break-all;\"> y = 2^x       x = 2^y\r<BR>\n" );
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document.write( "Evaluate the exponential equation for three positive values of x, three negative values of x, and x =0.  Transform the second expression into the equivalent logarithmic equation;  and evaluate the logarithmic equation for three values of x that ar greater than 1, three values of x that are between 0 and 1, and at x = 1.  Use the resulting ordered pairs to plot the graph of each function.\r<BR>\n" );
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document.write( "On the first equation I got y = 2^x\r<BR>\n" );
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document.write( "X   Y   3 positives of x<BR>\n" );
document.write( "1   2<BR>\n" );
document.write( "2   4<BR>\n" );
document.write( "3   8\r<BR>\n" );
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document.write( "X   Y   3 negatives of x<BR>\n" );
document.write( "-1  -2<BR>\n" );
document.write( "-2  -4<BR>\n" );
document.write( "-3  -8\r<BR>\n" );
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document.write( "X    Y  x = 0<BR>\n" );
document.write( "0    2\r<BR>\n" );
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document.write( "I do not under stand the second part of this equation. Is the first part correct?\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 #160876 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=160876\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> y = 2^x x = 2^y<BR>\n" );
document.write( "Evaluate the exponential equation for three positive values of x, three negative values of x, and x =0. Transform the second expression into the equivalent logarithmic equation; and evaluate the logarithmic equation for three values of x that ar greater than 1, three values of x that are between 0 and 1, and at x = 1. Use the resulting ordered pairs to plot the graph of each function.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "FIRST EQUATION<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2%5Ex ALIGN=MIDDLE  ALT=\"y+=+2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x is the exponent, 2 is the base.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "when x is positive:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E1+=+2 ALIGN=MIDDLE  ALT=\"2%5E1+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E2+=+4 ALIGN=MIDDLE  ALT=\"2%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E3+=+8 ALIGN=MIDDLE  ALT=\"2%5E3+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "when x is negative:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28-1%29+=+1%2F%282%5E1%29+=+1%2F2 ALIGN=MIDDLE  ALT=\"2%5E%28-1%29+=+1%2F%282%5E1%29+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28-2%29+=+1%2F%282%5E2%29+=+1%2F4 ALIGN=MIDDLE  ALT=\"2%5E%28-2%29+=+1%2F%282%5E2%29+=+1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28-3%29+=+1%2F%282%5E3%29+=+1%2F8 ALIGN=MIDDLE  ALT=\"2%5E%28-3%29+=+1%2F%282%5E3%29+=+1%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "graph of y = 2^x is shown below:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28300%2C300%2C-10%2C10%2C-10%2C10%2C2%5Ex%29 ALIGN=MIDDLE  ALT=\"graph+%28300%2C300%2C-10%2C10%2C-10%2C10%2C2%5Ex%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "SECOND EQUATION<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2%5Ey ALIGN=MIDDLE  ALT=\"x+=+2%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "y is the exponent, 2 is the base.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "when y is positive:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E1+=+2 ALIGN=MIDDLE  ALT=\"2%5E1+=+2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E2+=+4 ALIGN=MIDDLE  ALT=\"2%5E2+=+4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E3+=+8 ALIGN=MIDDLE  ALT=\"2%5E3+=+8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "when y is negative:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28-1%29+=+1%2F%282%5E1%29+=+1%2F2 ALIGN=MIDDLE  ALT=\"2%5E%28-1%29+=+1%2F%282%5E1%29+=+1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28-2%29+=+1%2F%282%5E2%29+=+1%2F4 ALIGN=MIDDLE  ALT=\"2%5E%28-2%29+=+1%2F%282%5E2%29+=+1%2F4\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E%28-3%29+=+1%2F%282%5E3%29+=+1%2F8 ALIGN=MIDDLE  ALT=\"2%5E%28-3%29+=+1%2F%282%5E3%29+=+1%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "You cannot graph this directly since the graphs are set up so that the independent variable is x and the dependent variable is y.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "In order to graph this, you have to solve for y.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "Your equation is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2%5Ey ALIGN=MIDDLE  ALT=\"x+=+2%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "The basic definition of an exponential equation is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+b%5Ex ALIGN=MIDDLE  ALT=\"y+=+b%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+log%28b%2Cy%29 ALIGN=MIDDLE  ALT=\"x+=+log%28b%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "When you reverse the x and y like we have in this second form of the exponential equation, then the basic definition becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+b%5Ey ALIGN=MIDDLE  ALT=\"x+=+b%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+log%28b%2Cx%29 ALIGN=MIDDLE  ALT=\"y+=+log%28b%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "----- <BR>\n" );
document.write( "Your equation is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2%5Ey ALIGN=MIDDLE  ALT=\"x+=+2%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "By the basic definition of exponents, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2%5Ey ALIGN=MIDDLE  ALT=\"x+=+2%5Ey\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+log%282%2Cx%29 ALIGN=MIDDLE  ALT=\"y+=+log%282%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "The graph of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+log%282%2Cx%29 ALIGN=MIDDLE  ALT=\"y+=+log%282%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is shown below:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28300%2C300%2C-10%2C10%2C-10%2C10%2Clog%282%2Cx%29%29 ALIGN=MIDDLE  ALT=\"graph+%28300%2C300%2C-10%2C10%2C-10%2C10%2Clog%282%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+log%282%2Cx%29 ALIGN=MIDDLE  ALT=\"y+=+log%282%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the inverse equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2%5Ex ALIGN=MIDDLE  ALT=\"y+=+2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You derive the inverse equation by doing exactly what they are asking you to do when they showed you the two equations.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "In order to find the inverse equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2%5Ex ALIGN=MIDDLE  ALT=\"y+=+2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > you would do the following:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "Solve for x:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "By the basic definition of exponents, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2%5Ex ALIGN=MIDDLE  ALT=\"y+=+2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+log%282%2Cy%29 ALIGN=MIDDLE  ALT=\"x+=+log%282%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "You have just solved for x by just applying the basic definition of exponents.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "You have:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+log%282%2Cy%29 ALIGN=MIDDLE  ALT=\"x+=+log%282%2Cy%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "Next you transpose the x and the y in the equation.<BR>\n" );
document.write( "Your equation becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+log%282%2Cx%29 ALIGN=MIDDLE  ALT=\"y+=+log%282%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+log%282%2Cx%29 ALIGN=MIDDLE  ALT=\"y+=+log%282%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is the inverse equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2%5Ex ALIGN=MIDDLE  ALT=\"y+=+2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "You have just derived the inverse equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2%5Ex ALIGN=MIDDLE  ALT=\"y+=+2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  That equation is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+log%282%2Cx%29 ALIGN=MIDDLE  ALT=\"y+=+log%282%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "The inverse equation is the reflection of the normal equation about the line y = x.<BR>\n" );
document.write( "The following graph shows the normal equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2%5Ex ALIGN=MIDDLE  ALT=\"y+=+2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the inverse equation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+log%282%2Cx%29 ALIGN=MIDDLE  ALT=\"y+=+log%282%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the line <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+x ALIGN=MIDDLE  ALT=\"y+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28300%2C300%2C-10%2C10%2C-10%2C10%2C2%5Ex%2Clog%282%2Cx%29%2Cx%29 ALIGN=MIDDLE  ALT=\"graph+%28300%2C300%2C-10%2C10%2C-10%2C10%2C2%5Ex%2Clog%282%2Cx%29%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "If you draw a perpendicular line anywhere through the line y = x, the intersection of that line through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2%5Ex ALIGN=MIDDLE  ALT=\"y+=+2%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be (a,b), and the intersection of that line through <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+log%282%2Cx%29 ALIGN=MIDDLE  ALT=\"y+=+log%282%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > will be (b,a).<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "for example:<BR>\n" );
document.write( "when x = 2, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+2%5E2 ALIGN=MIDDLE  ALT=\"y+=+2%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 4 so the coordinates are (2,4)<BR>\n" );
document.write( "when x = 4, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+log%282%2C4%29 ALIGN=MIDDLE  ALT=\"y+=+log%282%2C4%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > = 2 so the coordinates are (4,2)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "The equation of the line perpendicular to the line y = x and passing through the points (2,4) and (4,2) is y = -x + 6.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "Adding that line to the graph of the 3 equations yields the following:<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph+%28300%2C300%2C-10%2C10%2C-10%2C10%2C2%5Ex%2Clog%282%2Cx%29%2Cx%2C-x%2B6%29 ALIGN=MIDDLE  ALT=\"graph+%28300%2C300%2C-10%2C10%2C-10%2C10%2C2%5Ex%2Clog%282%2Cx%29%2Cx%2C-x%2B6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "As you can see, the two points intersecting with the line perpendicular to the line y = x are the same distance from that line making the two graphs symmetric to each other about the line y = x.<BR>\n" );
document.write( "-----<BR>\n" );
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document.write( "</TD></TR></TABLE>\n" );