document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=172962>Question 172962</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I would like to get some assistance with this question.  Give an example of an exponential function. Convert this exponential function to a logarithmic function. Plot the graph.<BR>\n" );
document.write( "thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #127816 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=127816\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the general form of an exponential function 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;\" ><BR>\n" );
document.write( "where:<BR>\n" );
document.write( "b is the base<BR>\n" );
document.write( "and<BR>\n" );
document.write( "x is the exponent<BR>\n" );
document.write( "and<BR>\n" );
document.write( "y is the result of the base raised to the exponent.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the general form of a logarithmic function is:<BR>\n" );
document.write( "<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( "where:<BR>\n" );
document.write( "b is the base<BR>\n" );
document.write( "and <BR>\n" );
document.write( "y is the exponent<BR>\n" );
document.write( "and<BR>\n" );
document.write( "x is the result of the base raised to the exponent.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the rule for logarithms 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( "conversely:<BR>\n" );
document.write( "<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;\" > if and only if: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+b%5Ey%29 ALIGN=MIDDLE  ALT=\"x+=+b%5Ey%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the exponential form of the equation and the logarithmic form of the equation are inverses of each other. <BR>\n" );
document.write( "here's an example:<BR>\n" );
document.write( "10^2 = 100<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the exponential form of this equation is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+10%5Ex ALIGN=MIDDLE  ALT=\"y+=+10%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 2<BR>\n" );
document.write( "y = 100<BR>\n" );
document.write( "coordinate point on the graph of this equation is: (2,100)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the logarithmic form of this equation is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+log%2810%2Cx%29 ALIGN=MIDDLE  ALT=\"y+=+log%2810%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "x = 100<BR>\n" );
document.write( "y = 2<BR>\n" );
document.write( "coordinate point on the graph of this equation is: (100,2)<BR>\n" );
document.write( "----<BR>\n" );
document.write( "in general:<BR>\n" );
document.write( "if f(x) = (a,b), and g(x) = (b,a) then the equations are inverses of each other.<BR>\n" );
document.write( "this is true for the two equations:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+f%28x%29+=+b%5Ex ALIGN=MIDDLE  ALT=\"y+=+f%28x%29+=+b%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+g%28x%29+=+log%28b%2Cx%29 ALIGN=MIDDLE  ALT=\"y+=+g%28x%29+=+log%28b%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "you can also graph these functions (i used the base of 10 - any base would work).<BR>\n" );
document.write( "the functions are:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+10%5Ex ALIGN=MIDDLE  ALT=\"y+=+10%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "and<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y+=+log%2810%2Cx%29 ALIGN=MIDDLE  ALT=\"y+=+log%2810%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we used base b = 10 so that it can be graphed easily and solved for easily on the calculator.  the base can be any real number > 0.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the line y = x is inserted in the middle of the graph to show more clearly that these functions are inverses of each other which shows up as their being reflections about the line y = x (kind of like mirror images if you think of the line y = x as the mirror).<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "look below the graph for further comments.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=graph%28800%2C800%2C-10%2C10%2C-10%2C10%2C10%5Ex%2Clog%2810%2Cx%29%2Cx%29 ALIGN=MIDDLE  ALT=\"graph%28800%2C800%2C-10%2C10%2C-10%2C10%2C10%5Ex%2Clog%2810%2Cx%29%2Cx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "a practical example:<BR>\n" );
document.write( "suppose you are given:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4500+=+10%5Ex ALIGN=MIDDLE  ALT=\"4500+=+10%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "how would you solve this?<BR>\n" );
document.write( "let y = 4500<BR>\n" );
document.write( "let x = x<BR>\n" );
document.write( "let b (the base) = 10<BR>\n" );
document.write( "using the laws of logarithms, you know that:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4500+=+10%5Ex ALIGN=MIDDLE  ALT=\"4500+=+10%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "if and only if:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+log%2810%2C4500%29 ALIGN=MIDDLE  ALT=\"x+=+log%2810%2C4500%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "so you go to the calculator and get the log of 4500 which is: 3.653212514<BR>\n" );
document.write( "how do you know this is good?<BR>\n" );
document.write( "you use the calculator again to take <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%5E3.653212514 ALIGN=MIDDLE  ALT=\"10%5E3.653212514\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get: 4500<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "what if the base is not 10?<BR>\n" );
document.write( "how to solve?<BR>\n" );
document.write( "there's a conversion formula you can use to convert any base to the base of 10 so you can solve it on your calculator.<BR>\n" );
document.write( "that conversion formula is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2Cx%29+=+log%2810%2Cx%29%2Flog%2810%2Cb%29 ALIGN=MIDDLE  ALT=\"log%28b%2Cx%29+=+log%2810%2Cx%29%2Flog%2810%2Cb%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "here's an example of how that works.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "your new problem:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=34561+=+3.5%5Ex ALIGN=MIDDLE  ALT=\"34561+=+3.5%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you know that this is true if and only if:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283.5%2C34561%29+=+x ALIGN=MIDDLE  ALT=\"log%283.5%2C34561%29+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you can't solve that on your calculator directly, but you can convert it to the base of 10 using the conversion formula.<BR>\n" );
document.write( "that would be:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283.5%2C34561%29+=+log%2810%2C34561%29%2Flog%2810%2C3.5%29+=+x ALIGN=MIDDLE  ALT=\"log%283.5%2C34561%29+=+log%2810%2C34561%29%2Flog%2810%2C3.5%29+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "-----<BR>\n" );
document.write( "you use your calculator to get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283.5%2C34561%29+=+log%2810%2C34561%29%2Flog%2810%2C3.5%29+=+%284.5385863%2F.544068044%29+=+x ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which becomes:<BR>\n" );
document.write( "x = 8.341946099<BR>\n" );
document.write( "this means that:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=34561+=+3.5%5E8.341946099 ALIGN=MIDDLE  ALT=\"34561+=+3.5%5E8.341946099\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "you can use your calculator to prove that this is true:<BR>\n" );
document.write( "34561 = 34561 proving that it is true.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );