document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=277975>Question 277975</A>: <I><SPAN STYLE=\"word-break: break-all;\"> PLEASE HELP!!<BR>\n" );
document.write( "If an earthquake has a magnitude of 4.2 on the Richter scale, what is the magnitude on the Richter scale of an earthquake that has an intensity 20 times greater?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #202320 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=202320\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the richter scale is based on the log of 10.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "richter scale of 0 is equivalent to the log of an earthquake with a magnitude of 10^0.<BR>\n" );
document.write( "richter scale of 1 is equivalent to the log of an earthquake with a magnitude of 10^1.<BR>\n" );
document.write( "richter scale of 2 is equivalent to the log of an earthquake with a magnitude of 10^2.<BR>\n" );
document.write( "richter scale of 3 is equivalent to the log of an earthquake with a magnitude of 10^3.<BR>\n" );
document.write( "richter scale of 4 is equivalent to the log of an earthquake with a magnitude of 10^4.<BR>\n" );
document.write( "richter scale of 5 is equivalent to the log of an earthquake with a magnitude of 10^5.<BR>\n" );
document.write( "richter scale of 6 is equivalent to the log of an earthquake with a magnitude of 10^6.<BR>\n" );
document.write( "richter scale of 7 is equivalent to the log of an earthquake with a magnitude of 10^7.<BR>\n" );
document.write( "richter scale of 8 is equivalent to the log of an earthquake with a magnitude of 10^8.<BR>\n" );
document.write( "richter scale of 9 is equivalent to the log of an earthquake with a magnitude of 10^9.<BR>\n" );
document.write( "richter scale of 10 is equivalent to the log of an earthquake with a magnitude of 10^10.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since the richter scale is based on the log to the base 10, this means that each succeeding number on the richter scale is 10 * bigger than the previous number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "richter scale of 0 = log(10^0) = log(1) = earthquake with a magnitude of 1<BR>\n" );
document.write( "richter scale of 1 = log(10^1) = log(10) = earthquake with a magnitude of 10<BR>\n" );
document.write( "richter scale of 2 = log(10^2) = log(100) = earthquake with a magnitude of 100<BR>\n" );
document.write( "richter scale of 3 = log(10^3) = log(1000) = earthquake with a magnitude of 1000<BR>\n" );
document.write( "etc.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "now to your problem.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "problem states that earthquake has a magnitude of 4.2 on the richter scale.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this means that the log of x to the base 10 = 4.2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "by the laws of logarithms, that is true if and only if:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 10^4.2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "now you can use your calculator to find 10^4.2 and your equation becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 15848.93192\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that means that an earthquake with a magnitude of 15848.93192 measures 4.2 on the richter scale.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "for an earthquake to be 20 * more powerful than that, x would have to be equal to 20 * 15848.93192 as shown below:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 20 * 15848.93192 = 316978.6385\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "so you have x = 316978.6385 and you take the log of that to the base 10 to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(316978.6385) = 5.501029996\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this is true because 10^5.501029996 = 316978.6385.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your answer is that an earthquake with an intensity of 20 times greater would measure 5.501029996 on the richter scale.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you could have solved this a little differently as shown below:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "if an earthquake measures 4.2 on the richter scale, this means that:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4.2 = log(x) where x is the magnitude of the earthquake.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "for the earthquake to be 20 times as powerful, then the new measurement on the richter scale becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = log(20*x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "by the laws of logarithms, this becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = log(20) + log(x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since log(x) = 4.2, this becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = log(20) + 4.2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "using your calculator to find log(20), this becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = 1.301029996 + 4.2 which becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = 5.501029996\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that's the same answer we got before.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "we originally found x and then took 20 times that number.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this time we did not find x, but used the laws of logarithms.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "either way we got the same answer.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the basic rules of logarithms are:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "y = log(b,x) if and only if b^y = x\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(a*x) = log(x) + log(a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(x/a) = log(x) - log(a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(x^a) = a * log(x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "either way, your answer is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "an earthquake with a magnitude 20 * greater than an earthquake that measures 4.2 on the richter scale is an earthquake that measures 5.501029996 on the richter scale.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );