document.write( "Question 604207: log square root of (x^2-1)=2
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Algebra.Com's Answer #381078 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"log%28%28sqrt%28x%5E2-1%29%29%29=2\"
\n" ); document.write( "Solving equations where the variable is in the argument (or base) of a logarithm usually starts with transforming the equation into one of the following forms:
\n" ); document.write( "log(expression) = other-expression
\n" ); document.write( "or
\n" ); document.write( "log(expression) = log(other-expression

\n" ); document.write( "Your equation is already in the first form! With the first form the next step is to rewrite the equation in exponential form. In general \"log%28a%2C+%28p%29%29+=+q\" is equivalent to \"a%5Eq+=+p\". Using this pattern on your equation we get:
\n" ); document.write( "\"10%5E2+=+sqrt%28x%5E2-1%29\"
\n" ); document.write( "which simplifies to:
\n" ); document.write( "\"100+=+sqrt%28x%5E2-1%29\"
\n" ); document.write( "Now we can solve for x. First let's get rid of the square root by squaring both sides:
\n" ); document.write( "\"10000+=+x%5E2-1\"
\n" ); document.write( "Since there is no \"x\" term we can just add 1:
\n" ); document.write( "\"10001+=+x%5E2\"
\n" ); document.write( "and find the square roots:
\n" ); document.write( "\"0+%2B-+sqrt%2810001%29+=+x\"
\n" ); document.write( "(NOTE: Algebra.com's software will not let me use a \"plus or minus\" symbol wiout something in front of it. This is why I put the zero in front. Mathematically the zero is unnecessary.)

\n" ); document.write( "With this problem we have two reasons a check is required, not optional:
  • Any equation where the variable is in the argument of a log must be checked to make sure that the arguments are all positive.
  • Any time both sides of an equation are squared you must check for extraneous solutions (solution that work in the squared equation but not in the original equation.

\n" ); document.write( "Use the original equation to check:
\n" ); document.write( "\"log%28%28sqrt%28x%5E2-1%29%29%29=2\"
\n" ); document.write( "Checking \"x+=+sqrt%2810001%29\"
\n" ); document.write( "\"log%28%28sqrt%28%28sqrt%2810001%29%29%5E2-1%29%29%29=2\"
\n" ); document.write( "Simplifying...
\n" ); document.write( "\"log%28%28sqrt%2810001-1%29%29%29=2\"
\n" ); document.write( "\"log%28%28sqrt%2810000%29%29%29=2\"
\n" ); document.write( "Since square roots are positive, we can already see that this solution passes the first check. But we should continue to make sure this solution is not extraneous.
\n" ); document.write( "\"log%28100%29%29%29=2\"
\n" ); document.write( "2 = 2 Check!

\n" ); document.write( "Checking \"x+=+-sqrt%2810001%29\"
\n" ); document.write( "\"log%28%28sqrt%28%28-sqrt%2810001%29%29%5E2-1%29%29%29=2\"
\n" ); document.write( "Simplifying...
\n" ); document.write( "\"log%28%28sqrt%2810001-1%29%29%29=2\"
\n" ); document.write( "\"log%28%28sqrt%2810000%29%29%29=2\"
\n" ); document.write( "\"log%28100%29%29%29=2\"
\n" ); document.write( "2 = 2 Check!

\n" ); document.write( "So both \"sqrt%2810001%29\" and \"-sqrt%2810001%29\" are solutions to your equation. \n" ); document.write( "
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