document.write( "Question 238168: log5(x+2)+log5(x-2)=3 \n" ); document.write( "
Algebra.Com's Answer #175164 by jsmallt9(3758)\"\" \"About 
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When you are solving for variables that are in the argument(s) of logarithms, you want to use Algebra and the properties of logarithms to transform the equation into the form:
\n" ); document.write( "log(expression-with-variables) = some-other-expression

\n" ); document.write( "Transforming your equation is not difficult. All we need to do is use the property, \"log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29\". This allows us to \"combine\" the sum of these two different logs into one. Your equation, after using this property, now looks like:
\n" ); document.write( "\"log%285%2C+%28%28x%2B2%29%28x-2%29%29%29+=+3\"
\n" ); document.write( "And we have the desired form. Now we will rewrite this in exponential form, using the fact that \"log%28a%2C+%28p%29%29+=+q\" is equivalent to \"p+=+a%5Eq\":
\n" ); document.write( "\"%28x%2B2%29%28x-2%29+=+5%5E3\"
\n" ); document.write( "Simplifying this we get:
\n" ); document.write( "\"x%5E2+-+4+=+125\"
\n" ); document.write( "Now we can add 4 to both sides:
\n" ); document.write( "\"x%5E2+=+129\"
\n" ); document.write( "Square root of both sides:
\n" ); document.write( "\"sqrt%28x%5E2%29+=+sqrt%28129%29\"
\n" ); document.write( "\"abs%28x%29+=+sqrt%28129%29\"
\n" ); document.write( "Solving this we get:
\n" ); document.write( "\"x+=+sqrt%28129%29\" or \"x+=+-sqrt%28129%29\"

\n" ); document.write( "With logarithm equations it is important to check your answers. (It's more than just a good idea.) What we check for is to make sure that none of the arguments of the logarithms become zero or negative.
\n" ); document.write( "Checking \"x+=+sqrt%28129%29\"
\n" ); document.write( "\"log%285%2C+%28sqrt%28129%29%2B2%29%29+%2B+log%285%2C+%28sqrt%28129%29-2%29%29+=+3\"
\n" ); document.write( "Since \"sqrt%28129%29+%3E+2\" both arguments are positive.

\n" ); document.write( "Checking \"x+=+-sqrt%28129%29\"
\n" ); document.write( "\"log%285%2C+%28-sqrt%28129%29%2B2%29%29+%2B+log%285%2C+%28-sqrt%28129%29-2%29%29+=+3\"
\n" ); document.write( "Both arguments are negative so we have to reject this solution. (Important: It was not because \"-sqrt%28129%29\" was negative that we rejected it. We rejected it because the logarithm arguments became negative when x was \"-sqrt%28129%29\" that we rejected it. In other problems it might work out that a negative value for x would be OK but a positive one would not. And sometimes both answers check and sometimes no answers work.)
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