document.write( "Question 277571: [inside brackets = Base]
\n" ); document.write( "(log[5](3x+10))-(3log[5](4))=2
\n" ); document.write( "Solve the equation.
\n" ); document.write( "Thanks. \n" ); document.write( "

Algebra.Com's Answer #202148 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"log%285%2C+%283x%2B10%29%29+-+3log%285%2C+%284%29%29=2\"$
\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 all 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 3 in front of the second 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 second log we get:
\n" ); document.write( " $\"log%285%2C+%283x%2B10%29%29+-+log%285%2C+%284%5E3%29%29=2\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"log%285%2C+%283x%2B10%29%29+-+log%285%2C+%2864%29%29=2\"$
\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( " $\"log%285%2C+%283x%2B10%29%29%2F64%29%29=2\"$
\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( " $\"%283x+%2B+10%29%2F64+=+5%5E2\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"%283x%2B10%29%2F64+=+25\"$
\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 64 to get rid of the fraction:
\n" ); document.write( "3x + 10 = 1600
\n" ); document.write( "Subtracting 10 from each side:
\n" ); document.write( "3x = 1590
\n" ); document.write( "Dividing by 3 we get:
\n" ); document.write( "x = 530

\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( " $\"log%285%2C+%283x%2B10%29%29+-+3log%285%2C+%284%29%29=2\"$
\n" ); document.write( "Checking x = 530:
\n" ); document.write( " $\"log%285%2C+%283%28530%29%2B10%29%29+-+3log%285%2C+%284%29%29=2\"$
\n" ); document.write( "which simplifies to
\n" ); document.write( " $\"log%285%2C+%281600%29%29+-+3log%285%2C+%284%29%29=2\"$
\n" ); document.write( "Both arguments are positive so it looks good. (You're welcome to finish the check on your own.)

\n" ); document.write( "The solution to your equation, then, is x = 530}}} \n" ); document.write( "

\n" );