document.write( "Question 277569: 2logx - log5 = 4 \n" ); document.write( "

Algebra.Com's Answer #202147 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"2log%28%28x%29%29+-+log%28%285%29%29+=+4\"$
\n" ); document.write( "We want the equation in the form:
\n" ); document.write( "log(expression) = other-expression
\n" ); document.write( "So somehow we need to combine the two logarithms into one. These two logarithms are not like terms so we cannot subtract them. But there is a property of logarithms, $\"log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29\"$ , which can be used to combine two logarithms if al of the following are true:

there is a \"-\" between them
the bases of the logarithms are the same
the coefficients of the logarithms are 1's

\n" ); document.write( "Your logarithms meet the first two but not the last. So now our goal is to get rid of the 2 in front of the first log. And fortunately there is another property of logarithms, $\"q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29\"$ , which can be used to move a coefficient of a logarithm into its argument as an exponent. Using this on your first log we get:
\n" ); document.write( " $\"log%28%28x%5E2%29%29+-+log%28%285%29%29+=+4\"$
\n" ); document.write( "These are still not like terms so we still cannot subtract them. But we can now use the other property to combine them:
\n" ); document.write( " $\"log%28%28x%5E2%2F5%29%29+=+4\"$
\n" ); document.write( "We now have the desired form. Once we have this form the next step is to rewrite the equation in exponential form:
\n" ); document.write( " $\"x%5E2%2F5+=+10%5E4\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"x%5E2%2F5+=+10000\"$
\n" ); document.write( "Now the variable is out of the argument where we can \"get at it\". Solving this for x we start by multiplying both sides by 5 to get rid of the fraction:
\n" ); document.write( " $\"x%5E2+=+50000\"$
\n" ); document.write( "So
\n" ); document.write( " $\"x+=+sqrt%2850000%29\"$ or $\"x+=+-sqrt%2850000%29\"$
\n" ); document.write( "Simplifying these square roots we get:
\n" ); document.write( " $\"x+=+100sqrt%285%29\"$ or $\"x+=+-100sqrt%285%29\"$

\n" ); document.write( "When solving logarithmic equations, it is important (not just a good idea) to check your answers. Even if we've done everything correct so far, we need to make sure that each answer makes the argument of any logarithms positive.

\n" ); document.write( "Always use the original equation to check:
\n" ); document.write( " $\"2log%28%28x%29%29+-+log%28%285%29%29+=+4\"$
\n" ); document.write( "Checking $\"x+=+100sqrt%285%29\"$
\n" ); document.write( " $\"2log%28%28100sqrt%285%29%29%29+-+log%28%285%29%29+=+4\"$
\n" ); document.write( "Both arguments are positive so it looks good. (You're welcome to finish the check on your own.)

\n" ); document.write( "Checking $\"x+=+-100sqrt%285%29\"$
\n" ); document.write( " $\"2log%28%28-100sqrt%285%29%29%29+-+log%28%285%29%29+=+4\"$
\n" ); document.write( "As you can see the argument to the first logarithm is negative. We cannot allow this to occur. So we reject this solution.

\n" ); document.write( "The only solution to your equation, then, is $\"x+=+100sqrt%285%29\"$ \n" ); document.write( "

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