document.write( "Question 406163: log base 9 of 6x cubed minus log base 9 of 2x equals 1 \n" ); document.write( "

Algebra.Com's Answer #286870 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"log%289%2C+%286x%5E3%29%29+-+log%289%2C+%282x%29%29+=+1\"$
\n" ); document.write( "Solving equations where the variable is in the argument 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( "With your \"non-log\" term of 1 it will be more difficult to achieve the second form. So we will aim for the first form. The first form has a single logarithm equal to some expression. Your equation has a difference of two logarithms equal to a 1. Somehow we need to combine the two logarithms into one logarithm.

\n" ); document.write( "We cannot just subtract the two logarithms. They are not like terms. Like logarithmic terms have the same base and the same argument. Your logarithms have the same base, 9, but different arguments, $\"6x%5E3\"$ and 2x.

\n" ); document.write( "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 into one when:

the logarithms are of the same base; and
the logarithms have coefficients of 1; and
there is a \"-\" between them.

\n" ); document.write( "Your two logarithms meet all three requirements so we can use this property to combine them into one:
\n" ); document.write( " $\"log%289%2C+%28%286x%5E3%29%2F%282x%29%29%29%29+=+1\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"log%289%2C+%283x%5E2%29%29+=+1\"$
\n" ); document.write( "We now have the first form. The next step with the first form is to rewrite the equation in exponential form. In general $\"log%28a%2C+%28p%29%29+=+q\"$ is equivalent to $\"p+=+a%5Eq\"$ . Using this pattern on your equation we get:
\n" ); document.write( " $\"3x%5E2+=+9%5E1\"$
\n" ); document.write( "which simplifies to:
\n" ); document.write( " $\"3x%5E2+=+9\"$
\n" ); document.write( "This equation we can solve. This is a quadratic so we want one side to be zero. Subtracting 9 from each side we get:
\n" ); document.write( " $\"3x%5E2+-+9+=+0\"$
\n" ); document.write( "The only factoring that can be done is to factor out a 3:
\n" ); document.write( " $\"3%28x%5E2+-+3%29+=+0\"$
\n" ); document.write( "You misght be able to \"see\" what values of x will make $\"x%5E2-3\"$ zero. If not, then use the Quadratic Formula:
\n" ); document.write( " $\"x+=+%28-%280%29+%2B-+sqrt%28%280%29%5E2+-+4%281%29%28-3%29%29%29%2F2%281%29\"$
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( " $\"x+=+%28-%280%29+%2B-+sqrt%280+-+4%281%29%28-3%29%29%29%2F2%281%29\"$
\n" ); document.write( " $\"x+=+%28-%280%29+%2B-+sqrt%280+%2B12%29%29%2F2%281%29\"$
\n" ); document.write( " $\"x+=+%28-%280%29+%2B-+sqrt%2812%29%29%2F2%281%29\"$
\n" ); document.write( " $\"x+=+%280+%2B-+sqrt%2812%29%29%2F2\"$
\n" ); document.write( "(Sorry about the unneeded 0. Algebra.com's formula software will not let me use the \"plus or minus\" symbol without something in front of it.)
\n" ); document.write( " $\"x+=+%280+%2B-+sqrt%284%2A3%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%280+%2B-+sqrt%284%29%2Asqrt%283%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%280+%2B-+2%2Asqrt%283%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%282%280+%2B-+sqrt%283%29%29%29%2F2\"$
\n" ); document.write( " $\"x+=+%28cross%282%29%280+%2B-+sqrt%283%29%29%29%2Fcross%282%29\"$
\n" ); document.write( " $\"x+=+0+%2B-+sqrt%283%29\"$
\n" ); document.write( "In long form this is:
\n" ); document.write( " $\"x+=+sqrt%283%29\"$ or $\"x+=+-sqrt%283%29\"$

\n" ); document.write( "When solving equations where the variable is in the argument of a logarithm you must check your answer. You must check to make sure your solution(s) make all arguments of a logarithm positive. Any \"solution\" that makes an argument zero or negative must be rejected. And these rejected \"solutions\" can happen even if no mistakes have been made! This is why you must check, even if you are very good at Math.

\n" ); document.write( "Always use the original equation to check:
\n" ); document.write( " $\"log%289%2C+%286x%5E3%29%29+-+log%289%2C+%282x%29%29+=+1\"$
\n" ); document.write( "Checking $\"x+=+sqrt%283%29\"$ :
\n" ); document.write( " $\"log%289%2C+%286%28sqrt%283%29%29%5E3%29%29+-+log%289%2C+%282%28sqrt%283%29%29%29%29+=+1\"$
\n" ); document.write( "We really don't have to go any farther. Since square roots are positive, we should be able to see that both arguments will be positive when $\"x+=+sqrt%283%29\"$ . For the first argument, a positive number cubed will still be positive and that positive number times 6 will still be positive. For the second argument 2 times a positive is positive. So there is no reason to reject this solution. (The rest of the check will tell us if we made a mistake. This part is optional and you are welcome to finish it if you like.)

\n" ); document.write( "Checking $\"x+=+-sqrt%283%29\"$ :
\n" ); document.write( " $\"log%289%2C+%286%28-sqrt%283%29%29%5E3%29%29+-+log%289%2C+%282%28-sqrt%283%29%29%29%29+=+1\"$
\n" ); document.write( "We really don't have to go any farther. For the first argument, a negative cubed is still negative and that negative times 6 will still be negative. No matter how the other argument works out, we must reject this \"solution\".

\n" ); document.write( "So the only solution to your equation is:
\n" ); document.write( " $\"x+=+sqrt%283%29\"$

\n" ); document.write( "Important: We did not reject $\"x+=+-sqrt%283%29\"$ because the x was negative. We rejected $\"x+=+-sqrt%283%29\"$ because it made at least one argument of a logarithm negative! If the equation had been
\n" ); document.write( " $\"log%289%2C+%28-6x%5E3%29%29+-+log%289%2C+%28-2x%29%29+=+1\"$
\n" ); document.write( "we would be rejecting
\n" ); document.write( " $\"x+=+sqrt%283%29\"$
\n" ); document.write( "and keeping
\n" ); document.write( " $\"x+=+-sqrt%283%29\"$

\n" ); document.write( "Also, some equations will have more than one solution and none of them get rejected. On the other hand, if every \"solution\" gets rejected, then there is no solution to the equation. \n" ); document.write( "

\n" );