document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1016891>Question 1016891</A>: <I><SPAN STYLE=\"word-break: break-all;\"> prove that log15(1+(log30)/(log15))+1/2(log16(1+(log7)/(log4)))-log6((log3)/(log6)+1+(log7)/(log6))=2 </SPAN>\n" );
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document.write( "Question:<BR>\n" );
document.write( "prove that <BR>\n" );
document.write( "log15(1+(log30)/(log15))+1/2(log16(1+(log7)/(log4)))-log6((log3)/(log6)+1+(log7)/(log6))=2<BR>\n" );
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document.write( "Solution:<BR>\n" );
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document.write( "We used the symbol E to stand for the value of the left-hand side, or<BR>\n" );
document.write( "E=log15(1+(log30)/(log15))+1/2(log16(1+(log7)/(log4)))-log6((log3)/(log6)+1+(log7)/(log6))<BR>\n" );
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document.write( "We need to use some laws of logarithm:<BR>\n" );
document.write( "log(a)+log(b)=log(ab)<BR>\n" );
document.write( "log(a)-log(b)=log(a/b)<BR>\n" );
document.write( "(1/2)log(a)=log(sqrt(a))<BR>\n" );
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document.write( "Also, note that log(x) implies log(x) to the base 10.<BR>\n" );
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document.write( "The idea of solution is to break up each term and simplify.<BR>\n" );
document.write( "We have:<BR>\n" );
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document.write( "log15(1+(log30)/(log15))=log(15)+log(30)=log(450).........(1)<BR>\n" );
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document.write( "1/2(log16(1+(log7)/(log4)))=log(sqrt(16))(1+log(7)/log(4))<BR>\n" );
document.write( "=log(4)(1+log(7)/log(4))<BR>\n" );
document.write( "=log(4)+log(7)<BR>\n" );
document.write( "=log(28).................................................(2)<BR>\n" );
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document.write( "-log6((log3)/(log6)+1+(log7)/(log6))<BR>\n" );
document.write( "=-(log(3)+log(6)+log(7)<BR>\n" );
document.write( "=-(log(3*6*7))<BR>\n" );
document.write( "=-log(126)...............................................(3)<BR>\n" );
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document.write( "Putting the three terms together we have:<BR>\n" );
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document.write( "E=E=log15(1+(log30)/(log15))+1/2(log16(1+(log7)/(log4)))-log6((log3)/(log6)+1+(log7)/(log6))<BR>\n" );
document.write( "=log(450)+log(28)-log(126)<BR>\n" );
document.write( "=log(450*28/126)<BR>\n" );
document.write( "=log(100)<BR>\n" );
document.write( "=log(10^2)<BR>\n" );
document.write( "=2  which is exactly the value of the right hand side, so proved.\r<BR>\n" );
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document.write( "  </SPAN>\n" );
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