document.write( "Question 270142: Express y in terms of x
\n" ); document.write( "log base 4 y=log base 2 x+log base 2 9 - log base 2 3 \n" ); document.write( "

Algebra.Com's Answer #197947 by jsmallt9(3758) $\"\"$ $\"About$

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\n" ); document.write( "To express y in terms of x we will need to get y out of the argument of the logarithm. And to get a variable out of the argument of a logarithm you usually start by transforming the equation into one of these forms:
\n" ); document.write( "log(variable-expression) = other-expression
\n" ); document.write( "or
\n" ); document.write( "log(variable-expression) = log(other-expression)
\n" ); document.write( "(with the bases of the logarithms the same in the second form).

\n" ); document.write( "To reach either of these forms from your equation we will need to have the same bases for each logarithm. So we need to:

Make the bases of the logarithms equal.
Transform the new equation into one of the forms above.
Solve the equation from step #2 for y.

\n" ); document.write( "Let's see this in action.
\n" ); document.write( "1. Same bases. Since there is only one logarithm of base 4 we will change it to match the base of the others (base 2). There is a formula for converting bases of logarithms. To change a base \"a\" logarithm to an expression with base \"b\" logarithms: $\"log%28a%2C+%28x%29%29+=+log%28b%2C+%28x%29%29%2Flog%28b%2C+%28a%29%29\"$ . To change $\"log%284%2C+%28y%29%29\"$ into an equivalent expression of base 2 logarithms: $\"log%284%2C+%28y%29%29+=+log%282%2C+%28y%29%29%2Flog%282%2C+%284%29%29\"$ . Substituting this base 2 expression in for the base 4 logarithm our equation becomes:
\n" ); document.write( "

\n" ); document.write( "Fortunately, since $\"2%5E2+=+4\"$ then $\"log%282%2C+%284%29%29+=+2\"$ . So our equation simplifies to:
\n" ); document.write( "

\n" ); document.write( "2. Transform the equation into one of the forms above. Since every term of our equation is a logarithm, I'll going to work towards the second form. This will require that we combine the three logarithms on the right into a single logarithm. Fortunately we have properties of logarithms which allow us to do just that:

$\"log%28a%2C+%28p%29%29+%2B+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Aq%29%29\"$
$\"log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29+=+log%28a%2C+%28p%2Fq%29%29\"$

\n" ); document.write( "Since there is a \"+\" between the first two logarithms we'll use the first property to combine them:
\n" ); document.write( " $\"log%282%2C+%28y%29%29%2F2+=+log%282%2C+%28x%2A9%29%29+-+log%282%2C+%283%29%29\"$
\n" ); document.write( "or
\n" ); document.write( " $\"log%282%2C+%28y%29%29%2F2+=+log%282%2C+%289x%29%29+-+log%282%2C+%283%29%29\"$
\n" ); document.write( "Since there is a \"-\" between the remaing two logarithms, I'll use the second property to combine them:
\n" ); document.write( " $\"log%282%2C+%28y%29%29%2F2+=+log%282%2C+%289x%2F3%29%29\"$
\n" ); document.write( "which simplifies to
\n" ); document.write( " $\"log%282%2C+%28y%29%29%2F2+=+log%282%2C+%283x%29%29\"$
\n" ); document.write( "We are close to the second form. But the 2 in the denominator needs to go. If we multiply both sides by 2 we get:
\n" ); document.write( " $\"log%282%2C+%28y%29%29+=+2%2Alog%282%2C+%283x%29%29\"$
\n" ); document.write( "The 2 is still in the way but it is in a better place. Now we can use a third property of logarithms, $\"q%2Alog%28a%2C+%28p%29%29+=+log%28a%2C+%28p%5Eq%29%29\"$ , which allow us to move a coefficient into the argument as an exponent:
\n" ); document.write( " $\"log%282%2C+%28y%29%29+=+log%282%2C+%28%283x%29%5E2%29%29\"$
\n" ); document.write( "which simplifies to
\n" ); document.write( " $\"log%282%2C+%28y%29%29+=+log%282%2C+%289x%5E2%29%29\"$
\n" ); document.write( "We have finally achieved the second form (log(...) = log(...)).

\n" ); document.write( "3. Solve the equation for y. This requires that we get y out of the argument on the left. With the second form this is pretty easy. The equation says that the base 2 log of y is equal to the base 2 log of $\"9x%5E2\"$ . If their logs are equal then they are equal. In other words:
\n" ); document.write( " $\"y+=+9x%5E2\"$
\n" ); document.write( "which expresses y in terms of x. \n" ); document.write( "

\n" );