Question

Use the properties of logarithms to expand the following expression as much as possible. Simplify any numerical expressions that can be evaluated without a calculator.

log4(2x2 - 20x + 12y)

Answer 1

`f(x,y) = \log_(4) (x-5-√(25-6\cdot y))+\log_(4) (x-5+√(25-6\cdot y))`

Explanation:

Let be
`f(x,y) = \log_(4)(2\cdot x^(2)-20\cdot x +12\cdot y)`, this expression is simplified by algebraic and trascendental means. As first step, the second order polynomial is simplified. Its roots are determined by the Quadratic Formula, that is to say:

`r_(1,2) = \frac{20\pm \sqrt{(-20)^(2)-4\cdot (2)\cdot (12\cdot y)}}{2\cdot (2)}`

`r_(1,2) = 5\pm √(25-6\cdot y)`

The polynomial in factorized form is:

`(x-5-√(25-6\cdot y))\cdot (x-5+√(25-6\cdot y))`

The function can be rewritten and simplified as follows:

`f(x,y) = \log_(4) [(x-5-√(25-6\cdot y))\cdot (x-5+√(25-6\cdot y))]`

`f(x,y) = \log_(4) (x-5-√(25-6\cdot y))+\log_(4) (x-5+√(25-6\cdot y))`