SOLUTION: Express y in terms of x log base 4 y=log base 2 x+log base 2 9 - log base 2 3

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Question 270142: Express y in terms of x
log base 4 y=log base 2 x+log base 2 9 - log base 2 3
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!

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:
log(variable-expression) = other-expression
or
log(variable-expression) = log(other-expression)
(with the bases of the logarithms the same in the second form).

To reach either of these forms from your equation we will need to have the same bases for each logarithm. So we need to:
  1. Make the bases of the logarithms equal.
  2. Transform the new equation into one of the forms above.
  3. Solve the equation from step #2 for y.

Let's see this in action.
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:

Fortunately, since 2%5E2+=+4 then log%282%2C+%284%29%29+=+2. So our equation simplifies to:


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

Since there is a "+" between the first two logarithms we'll use the first property to combine them:
log%282%2C+%28y%29%29%2F2+=+log%282%2C+%28x%2A9%29%29+-+log%282%2C+%283%29%29
or
log%282%2C+%28y%29%29%2F2+=+log%282%2C+%289x%29%29+-+log%282%2C+%283%29%29
Since there is a "-" between the remaing two logarithms, I'll use the second property to combine them:
log%282%2C+%28y%29%29%2F2+=+log%282%2C+%289x%2F3%29%29
which simplifies to
log%282%2C+%28y%29%29%2F2+=+log%282%2C+%283x%29%29
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:
log%282%2C+%28y%29%29+=+2%2Alog%282%2C+%283x%29%29
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:
log%282%2C+%28y%29%29+=+log%282%2C+%28%283x%29%5E2%29%29
which simplifies to
log%282%2C+%28y%29%29+=+log%282%2C+%289x%5E2%29%29
We have finally achieved the second form (log(...) = log(...)).

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:
y+=+9x%5E2
which expresses y in terms of x.