document.write( "Question 604266: Solve for x: log base 5 (x^2+4x+4)=2 \n" ); document.write( "

Algebra.Com's Answer #381072 by jsmallt9(3759) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"log%285%2C+%28x%5E2%2B4x%2B4%29%29=2\"$
\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( " $\"5%5E2+=+x%5E2%2B4x+%2B4\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"25+=+x%5E2%2B4x+%2B4\"$
\n" ); document.write( "Now that x is out of the logarithm we can solve for it. Since this is a quadratic equation (because of the $\"x%5E2\"$ ) we want one side to be zero. Subtracting 25 from each side we get:
\n" ); document.write( " $\"0+=+x%5E2%2B4x+-21\"$
\n" ); document.write( "Now we factor (or use the Quadratic Formula). This factors fairly easily. The factors of -21 that add up to 4 are 7 and -3 so we get these factors:
\n" ); document.write( "0 = (x + 7)(x - 3)
\n" ); document.write( "From the Zero Product Property we know that one (or more) of these factors must be zero., So:
\n" ); document.write( "x + 7 = 0 or x - 3 = 0
\n" ); document.write( "Solving these we get:
\n" ); document.write( "x = -7 or x = 3

\n" ); document.write( "Checking solutions for equations like this is not optional. You must make sure that the proposed solutions make all arguments (and bases) are positive. Use the original equation to check:
\n" ); document.write( " $\"log%285%2C+%28x%5E2%2B4x%2B4%29%29=2\"$
\n" ); document.write( "Checking x = -7:
\n" ); document.write( " $\"log%285%2C+%28%28-7%29%5E2%2B4%28-7%29%2B4%29%29=2\"$
\n" ); document.write( " $\"log%285%2C+%2849%2B4%28-7%29%2B4%29%29=2\"$
\n" ); document.write( " $\"log%285%2C+%2849%2B%28-28%29%2B4%29%29=2\"$
\n" ); document.write( " $\"log%285%2C+%2825%29%29=2\"$
\n" ); document.write( "We can now see that the argument is positive. The required part of the check is complete. x = -7 is a solution.

\n" ); document.write( "Checking x = 3:
\n" ); document.write( " $\"log%285%2C+%28%283%29%5E2%2B4%283%29%2B4%29%29=2\"$
\n" ); document.write( " $\"log%285%2C+%289%2B4%283%29%2B4%29%29=2\"$
\n" ); document.write( " $\"log%285%2C+%289%2B12%2B4%29%29=2\"$
\n" ); document.write( " $\"log%285%2C+%2825%29%29=2\"$
\n" ); document.write( "Again we have a positive argument. So x = 3 is another solution to your equation.

\n" ); document.write( "NOTE: If either of our \"solutions\" had made an argument zero or negative, we would have to reject that solution. If all solutions get rejected then it means that there are no solutions to the equation (i.e. the equation was impossible from the start). \n" ); document.write( "

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