document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=380913>Question 380913</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve the following exponential functions<BR>\n" );
document.write( "a. 5 (with an exponent of) 5x+1 = 125<BR>\n" );
document.write( "b. 3 (with an exponent of x squared) = 81 (with an exponent of x) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #270378 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=270378\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> With many equations where the variable is in an exponent, one usually uses logarithms to solve the equation. An exception is when both sides of the equation can be expressed as powers of the same number. Both of your equations fit into this later category.<br><BR>\n" );
document.write( "a. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E%285x%2B1%29+=+125 ALIGN=MIDDLE  ALT=\"5%5E%285x%2B1%29+=+125\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=125+=+5%5E3 ALIGN=MIDDLE  ALT=\"125+=+5%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can write this as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5E%285x%2B1%29+=+5%5E3 ALIGN=MIDDLE  ALT=\"5%5E%285x%2B1%29+=+5%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Both sides are not powers of the same number, 5. In order for these powers of 5 to be equal, the exponents must be equal:<BR>\n" );
document.write( "5x+1 = 3<BR>\n" );
document.write( "This is a simple equation to solve:<BR>\n" );
document.write( "5x = 2<BR>\n" );
document.write( "x = 2/5<br><BR>\n" );
document.write( "P.S. As you noted in your thank-you note, I overlooked the exponent on the 81 in the second problem. Here's the corrected solution:<BR>\n" );
document.write( "b. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E%28%28x%5E2%29%29+=+81%5Ex ALIGN=MIDDLE  ALT=\"3%5E%28%28x%5E2%29%29+=+81%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Since <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=81+=+3%5E4 ALIGN=MIDDLE  ALT=\"81+=+3%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > we can rewrite this as:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E%28%28x%5E2%29%29+=+%283%5E4%29%5Ex ALIGN=MIDDLE  ALT=\"3%5E%28%28x%5E2%29%29+=+%283%5E4%29%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Using the power of a power rule for exponents (i.e. multiply) we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%5E%28%28x%5E2%29%29+=+3%5E%284x%29 ALIGN=MIDDLE  ALT=\"3%5E%28%28x%5E2%29%29+=+3%5E%284x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Again this equation is true only if the exponents are equal:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+=+4x ALIGN=MIDDLE  ALT=\"x%5E2+=+4x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is a quadratic equation. To solve it we need one side equal to zero. Subtracting 4x from each side we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+-+4x+=+0 ALIGN=MIDDLE  ALT=\"x%5E2+-+4x+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we factor (or use the Quadratic Formula). This factors very simply:<BR>\n" );
document.write( "x(x-4) = 0<BR>\n" );
document.write( "From the zero Product Property we know that this (or any) product can be zero <i>only</i> if one (or more) of the factors is zero. So:<BR>\n" );
document.write( "x = 0 or x-4 = 0<BR>\n" );
document.write( "Solving the second equation we get:<BR>\n" );
document.write( "x = 0 or x = 4 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );