document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1012454>Question 1012454</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Good evening, I have some difficulties with this logarithmic equation:<BR>\n" );
document.write( "4log4((x+1)^(1/2)) - log2(5-7x)=0<BR>\n" );
document.write( "I tried to solve changing the base in such a way:<BR>\n" );
document.write( "4log4((x+1)^(1/2)) = (log4(5-7x))/log4(2)<BR>\n" );
document.write( "but I do wrong somewhere solving the arguments.<BR>\n" );
document.write( "Can somebody help me explaining the right method?<BR>\n" );
document.write( "Many thanks  <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 #628399 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=628399\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i think i have it.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "your problem is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4log4((x+1)^(1/2)) - log2(5-7x) = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "add log2(5-7x) to both sides of the equation to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4log4((x+1)^.5) = log2(5-7x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can write it this way because 1/2 = .5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since alog(b) = log(b^a), then:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4log4((x+1)^.5) = log2(5-7x) becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log(((x+1)^.5)^4) = log2(5-7x) which becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log((x+1)^(.5*4)) = log2(5-7x) which becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log4((x+1)^2) = log2(5-7x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "change of base formula can be applied here.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "change of base formula says loga(x) = logb(x)/logb(a)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "using that formula, we change the base of 4 to base of 2 in the following manner.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log4((x+1)^2) = log2((x+1)^2)/log2(4)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "now log2(4) = y if and only if 2^y = 4.<BR>\n" );
document.write( "since 2^2 = 4, this means that y is equal to 2 and we get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log2(4) = 2, and ...,\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the formula of log4((x+1)^2) = log2((x+1)^2)/log2(4) becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log4((x+1)^2) = log2((x+1)^2)/2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "we now know that log4((x+1)^2) is equivalent to log2((x+1)^2)/2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "we go back to our formula of log4((x+1)^2) = log2(5-7x) and replace log4((x+1)^2) with log2((x+1)^2)/2 to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log2((x+1)^2)/2 = log2(5-7x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "we multiply both sides of this equation by 2 to get:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log2((x+1)^2) = 2log2(5-7x)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "since alog(b) = log(b^a), then log2((x+1)^2) = 2log2(5-7x) becomes:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "log2((x+1)^2) = log2((5-7x)^2)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "this is true if and only if (x+1)^2 = (5-7x)^2\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "solving this equation for x, we get x = 1 or x = 1/2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "to confirm whether these solutions are good, we have to go back to the original equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the original equation is:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "4log4((x+1)^(1/2)) - log2(5-7x) = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 1 is no good because then log2(5-7x) is the log of a negative number which is not allowed.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "the solution has to be x = 1/2 or we have no solution.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "x = 1/2 is a valid solution, so all we have to do now is confirm that is makes the original equation a true equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can confirm by replacing x with 1/2 and evaluating the original equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you will find that the equation becomes 0 when x = 1/2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can also graph the original equation.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "you can see that the graph of the equation is equal to 0 when x = .5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "that graph is shown below:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" src = \"http://theo.x10hosting.com/2016/010502.jpg\" alt=\"$$$\" </>\r<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" );