document.write( "Question 1116776: log(x+3)+ logx =1 solve x and show that the above equation will have an irrational root if the base is changed to 2, and the a rational root if the base is changed to 4 \n" ); document.write( "
Algebra.Com's Answer #731696 by rothauserc(4718)\"\" \"About 
You can put this solution on YOUR website!
case 1 is log(x+3)+ log (x) = 1 with base 2
\n" ); document.write( ":
\n" ); document.write( "(log(x)/log(2)) + (log(x+3)/log(2)) = 1
\n" ); document.write( ":
\n" ); document.write( "log(x) + log(x+3) = log(2)
\n" ); document.write( ":
\n" ); document.write( "log(x(x+3)) = log(2)
\n" ); document.write( ":
\n" ); document.write( "x(x+3) = 2
\n" ); document.write( ":
\n" ); document.write( "x^2 +3x = 2
\n" ); document.write( ":
\n" ); document.write( "complete the square
\n" ); document.write( ":
\n" ); document.write( "x^2 +3x +9/4 = 2 + 9/4 = 17/4
\n" ); document.write( ":
\n" ); document.write( "(x+(3/2))^2 = 17/4
\n" ); document.write( ":
\n" ); document.write( "x+(3/2) = square root(17)/2
\n" ); document.write( ":
\n" ); document.write( "x = square root(17)/2 -3/2
\n" ); document.write( ":
\n" ); document.write( "Note that this is the only solution for this case, the solution x = -square root(17)/2 -3/2 does not solve the equation(hint substitute for x)
\n" ); document.write( ":
\n" ); document.write( "case 2 is log(x+3)+ log (x) = 1 with base 4
\n" ); document.write( ":
\n" ); document.write( "x^2 +3x = 4
\n" ); document.write( ":
\n" ); document.write( "x^2 +3x -4 = 0
\n" ); document.write( ":
\n" ); document.write( "(x+4) * (x-1) = 0
\n" ); document.write( ":
\n" ); document.write( "x = -4, x = 1
\n" ); document.write( ":
\n" ); document.write( "x = 1 is the solution since logarithm of a negative number is a complex number
\n" ); document.write( ":
\n" ); document.write( " \n" ); document.write( "
\n" );