document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=713038>Question 713038</A>: <I><SPAN STYLE=\"word-break: break-all;\"> how do i find the exact value and the approximation of<BR>\n" );
document.write( "log7 10? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #438502 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=438502\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm assuming that the expression is:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%287%2C+%2810%29%29 ALIGN=MIDDLE  ALT=\"log%287%2C+%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If I'm wrong then you'll have to re-post your question making the expression more understandable.<br><BR>\n" );
document.write( "Did you post the problem exactly as it was given to you? I ask because it is impossible to find the exact value/number that equals <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%287%2C+%2810%29%29 ALIGN=MIDDLE  ALT=\"log%287%2C+%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If it was possible to find the exact number for logarithms like this we would never have to learn how to work with this syntax. Why would we ever use such cumbersome syntax as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > if we could replace them with numbers?!<br><BR>\n" );
document.write( "The best we can do for something <i>exactly</i> equal to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%287%2C+%2810%29%29 ALIGN=MIDDLE  ALT=\"log%287%2C+%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is to rewrite it in terms of logarithms of other bases. We can use the base conversion formula, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%29%29+=+log%28b%2C+%28p%29%29%2Flog%28b%2C+%28a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to change this base 7 log into an expression involving any other base.<br><BR>\n" );
document.write( "For two reasons I am going to use the change of base formula to change the base 7 log into an expression of base 10 logs:<ul><li>The 10 in the base 7 log</li><li>Our calculators \"know\" base 10 logs so we will be able to use our result to find the decimal approximation.</li></ul>Using the base conversion formula to convert<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%287%2C+%2810%29%29 ALIGN=MIDDLE  ALT=\"log%287%2C+%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "into an expression of base 10 logs we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28%2810%29%29%2Flog%28%287%29%29 ALIGN=MIDDLE  ALT=\"log%28%2810%29%29%2Flog%28%287%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "By definition, any log whose base matches its argument is equal to 1. So log(10) = 1 and our fraction simplifies to:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Flog%28%287%29%29 ALIGN=MIDDLE  ALT=\"1%2Flog%28%287%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is another exact expression (but not a \"value\") for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%287%2C+%2810%29%29 ALIGN=MIDDLE  ALT=\"log%287%2C+%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<br><BR>\n" );
document.write( "We can use this expression, and our calculators, to find a decimal approximation:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F0.84509804001425683071221625859264 ALIGN=MIDDLE  ALT=\"1%2F0.84509804001425683071221625859264\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to:<BR>\n" );
document.write( "1.1832946624549383268179285616469<br><BR>\n" );
document.write( "Note 1: No matter how many decimal places we use, we will never get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%287%2C+%2810%29%29 ALIGN=MIDDLE  ALT=\"log%287%2C+%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > exactly right. Even 1.1832946624549383268179285616469 is just a very, very close, rounded-off approximation of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%287%2C+%2810%29%29 ALIGN=MIDDLE  ALT=\"log%287%2C+%2810%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "Note 2: Since they're not exact anyway, feel free to round off these long decimals. </SPAN>\n" );
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